Quivers with potentials and their representations I: Mutations
QUIVERS WITH POTENTIALS AND THEIR
REPRESENTATIONS I: MUTATIONS
HARM DERKSEN, JERZY WEYMAN, AND ANDREI ZELEVINSKY
Abstract. We study quivers with relations given by non-commutative analogs of
Jacobian ideals in the complete path algebra. This framework allows us to give
a representation-theoretic interpretation of quiver mutations at arbitrary vertices.
This gives a far-reaching generalization of Bernstein-Gelfand-Ponomarev reflection
functors. The motivations for this work come from several sources: superpotentials
in physics, Calabi-Yau algebras, cluster algebras.
Contents
1. Introduction 1
2. Quivers and path algebras 5
3. Potentials and their Jacobian ideals 8
4. Quivers with potentials 12
5. Mutations of quivers with potentials 20
6. Some mutation invariants 25
7. Nondegenerate QPs 29
8. Rigid QPs 32
9. Relation to cluster-tilted algebras 36
10. Decorated representations and their mutations 39
11. Some three-vertex examples 50
12. Some open problems 54
13. Appendix. Proof of Lemma 4.12 55
Acknowledgments 57
References 57
1. Introduction
The main objects of study in this paper are quivers with potentials (QPs for short).
Roughly speaking, a QP is a quiver Q together with an element S of the path algebra
of Q such that S is a linear combination of cyclic paths. We associate to S the two-
sided ideal J(S) in the path algebra generated by the (noncommutative) partial
derivatives of S with respect to the arrows of Q. We refer to J(S) as the Jacobian
Date: April 18, 2007; revised July 10, 2007 and March 12, 2008.
2000 Mathematics Subject Classification. Primary 16G10, Secondary 16G20, 16S38.
Research of H. D. supported by the NSF grant DMS-0349019. Research of J. W. supported by
the NSF grant DMS-0600229. Research of A. Z. supported by the NSF grant DMS-0500534 and
by a Humboldt Research Award.
http://arxiv.org/abs/0704.0649v4
2 HARM DERKSEN, JERZY WEYMAN, AND ANDREI ZELEVINSKY
ideal, and to the quotient of the path algebra modulo J(S) as the Jacobian algebra.
They appeared in physicists’ work on superpotentials in the context of the Seiberg
duality in mirror symmetry (see e.g., [17, 2, 6]). Since in some of their work the
superpotentials are required to satisfy some form of Serre duality, we prefer not to
use this terminology, and just refer to S as a potential ; another reason for this is
that we are working with the completed path algebra, so our potentials are possibly
infinite linear combinations of cyclic paths. The Jacobian algebras also play an
important role in the recent work on Calabi-Yau algebras [4, 25, 26, 27].
In this paper we introduce and study mutations for QPs and their (decorated)
representations. In the context of Calabi-Yau algebras, the mutations were discussed
in [26] but our approach is much more elementary and down-to-earth. Namely, we
develop the setup that directly extends to QPs the Bernstein-Gelfand-Ponomarev
reflection functors [3] and their “decorated” version [28].
The original motivation for our study comes from the theory of cluster algebras
introduced and studied in a series of papers [18, 19, 1, 20]. In this paper, we deal only
with the underlying combinatorics of this theory embodied in skew-symmetrizable
integer matrices and their mutations. Furthermore, we restrict our attention to skew-
symmetric integer matrices. Such matrices can be encoded by quivers without loops
and oriented 2-cycles. Namely, a skew-symmetric integer n × n matrix B = (bi,j)
corresponds to a quiver Q(B) with vertices 1, . . . , n, and bi,j arrows from j to i
whenever bi,j > 0. For every vertex k, the mutation at k transforms B into another
skew-symmetric integer n × n matrix µk(B) = B = (bi,j). The formula for bi,j is
given below in (7.4). It is well-known (see Proposition 7.1 below) that the quiver
Q(B) can be obtained from Q(B) by the following three-step procedure:
Step 1. For every incoming arrow a : j → k and every outgoing arrow b : k → i,
create a “composite” arrow [ba] : j → i; thus, whenever bi,k, bk,j > 0, we
create bi,kbk,j new arrows from j to i.
Step 2. Reverse all arrows at k; that is, replace each arrow a : j → k with a⋆ : k → j,
and b : k → i with b⋆ : i→ k.
Step 3. Remove any maximal disjoint collection of oriented 2-cycles (that can appear
as a result of creating new arrows in Step 1).
In the case where k is a source or a sink of Q(B), the first and last steps of the
above procedure are not applicable, so Q(B) is obtained from Q(B) by just reversing
all the arrows at k. In this situation, J.Bernstein, I. Gelfand, and V. Ponomarev [3]
introduced the reflection functor at k sending representations of a quiver Q(B) (with-
out relations) into representations of Q(B). A modification of these functors acting
on decorated representations was introduced in [28] to establish a link between clus-
ter algebras and quiver representations (the definition of decorated representations
for general QPs is given below in Section 10).
The elementary approach of [28] has not been further pursued until now, giving
way to a more sophisticated approach via cluster categories and cluster-tilted alge-
bras developed in [7, 8, 9, 10, 12, 13, 14, 15] and many other publications. Most of
the results in these papers are for the quivers obtained by mutations from hereditary
algebras (i.e., quivers without oriented cycles and without relations). In this paper
we return to the more elementary point of view of [28] and propose an alternative
QUIVERS WITH POTENTIALS I 3
approach (which is in fact more general, since we do not impose any restrictions
on quivers in question). In this approach, the mutations at arbitrary vertices (not
just sources or sinks) are defined for QPs and their decorated representations. The
construction for QPs is carried out in Section 5, and for their representations in Sec-
tion 10. It turns out to be rather delicate and requires a lot of technical preparation.
The first two steps of the above mutation procedure extend to QPs in a relatively
straightforward way, but Step 3 presents a real challenge: we need to accompany
the removal of oriented 2-cycles from a quiver with a suitable modification of the
potential, leaving the corresponding Jacobian algebra unchanged. Our main device
in dealing with this difficulty is Theorem 4.6, which is the crucial technical result
of the paper. Roughly speaking, Theorem 4.6 asserts that every potential S can be
transformed by an automorphism of the path algebra into the sum of two potentials
Striv and Sred on the disjoint sets of arrows, where the trivial part Striv is a linear
combination of cyclic 2-paths, while the reduced part Sred involves only cyclic d-paths
with d ≥ 3. Furthermore, the Jacobian algebra of Sred is isomorphic to that of S.
Several comments on this result are in order. First, our arguments heavily de-
pend on the setup using completed path algebras, thus allowing potentials to involve
infinite sums of cyclic paths. Second, the reduction S 7→ Sred is not given by a
canonical procedure. As a consequence, our construction of mutations for QPs and
their representations is not functorial in any obvious sense. On the positive side, we
prove that every mutation is a well-defined transformation on the right-equivalence
classes of QPs (and their representations), where, roughly speaking, two QPs are
right-equivalent if they can be obtained from each other by an automorphism of the
path algebra (for more precise definitions see Definitions 4.2 and 10.2).
Finally, it is important to keep in mind that, even with the help of Theorem 4.6,
in order to get rid of all oriented 2-cycles in the mutated QP, one needs to impose
some “genericity” conditions on the initial potential S. These conditions are studied
in Section 7. They are not very explicit in general, but we introduce an important
class or rigid QPs (see Definitions 6.7 and 6.10) for which the absence of oriented
2-cycles after any sequence of QP mutation is guaranteed.
We now describe the contents of the paper in more detail. In Section 2 we introduce
an algebraic setup for dealing with quivers and their path algebras. We fix a base field
K, and encode a quiver with the vertex set Q0 and the arrow set Q1 by its vertex span
R = KQ0 and arrow span A = KQ1 . Thus, R is a finite-dimensional commutative
K-algebra, and A is a finite-dimensional R-bimodule. We then introduce the path
algebra
R〈A〉 =
and, more importantly for our purposes, the complete path algebra
R〈〈A〉〉 =
here Ad stands for the d-fold tensor power of A as an R-bimodule. We view R〈〈A〉〉
as a topological algebra via the m-adic topology, where m is the two-sided ideal
generated by A.
4 HARM DERKSEN, JERZY WEYMAN, AND ANDREI ZELEVINSKY
In Section 3 we introduce some of our main objects of study: potentials and
their Jacobian ideals. It is natural to view potentials as elements of the trace space
R〈〈A〉〉/{R〈〈A〉〉, R〈〈A〉〉}, where {R〈〈A〉〉, R〈〈A〉〉} is the closure of the vector sub-
space in R〈〈A〉〉 spanned by all commutators. It is more convenient for us to define
a potential S as an element of the cyclic part of R〈〈A〉〉; for all practical purposes,
S can be replaced by a cyclically equivalent potential, that is, the one with the same
image in the trace space. To define the Jacobian ideal J(S) and derive its basic
properties, we develop the formalism of cyclic derivatives, in particular, establishing
“cyclic” versions of the Leibniz rule and the chain rule. The main result of Section 3
is Proposition 3.7 that asserts that any isomorphism ϕ of path algebras sends J(S)
to J(ϕ(S)). Note that cyclic derivatives for general non-commutative algebras were
introduced in [29], and the results we present can be easily deduced from those in
loc.cit. For the convenience of the reader, we present complete independent proofs.
Victor Ginzburg informed us that in the context of path algebras of quivers, cyclic
derivatives were introduced and studied in [5, 24], and that Proposition 3.7 is a
consequence of the geometric interpretation of J(S) given in [25, Definition 5.1.1,
Lemma 5.1.3].
In Section 4 we introduce quivers with potentials (QPs) and define the right-
equivalence relation on them, which plays an important role in the paper. We then
state and prove the key technical result of the paper: Splitting Theorem 4.6, already
discussed above. The proof is elementary but pretty involved; it uses in an essential
way the topology in a complete path algebra. In order not to interrupt the argument,
we move to the Appendix our treatment of the topological properties needed for the
proof of one of the technical lemmas.
In Section 5 we finally introduce the mutations of QPs. Using Theorem 4.6, we
prove that the mutation at an arbitrary vertex is a well-defined involution on the set
of right-equivalence classes of reduced QPs (Theorem 5.7).
In Section 6, we study some mutation invariants of QPs. In particular, we show
that mutations preserve the class of QPs with finite-dimensional Jacobian algebras
(Corollary 6.6). Another important property of QPs preserved by mutations is rigid-
ity (Corollary 6.11), which was already mentioned above. For the precise definition
of rigid QPs see Definitions 6.7 and 6.10 below; intuitively, a QP is rigid if its right-
equivalence class is invariant under infinitesimal deformations.
In Section 7, we introduce and study nondegenerate QPs, that is, those to which
one can apply an arbitrary sequence of mutations without creating oriented 2-cycles.
In Corollary 7.4 we show that nondegeneracy is guaranteed by non-vanishing of
countably many nonzero polynomial functions on the space of potentials. In par-
ticular, if the base field K is uncountable, a nondegenerate QP exists for every
underlying quiver.
Section 8 contains some examples of rigid and non-rigid potentials and some further
results illustrating the importance of rigidity. A simple but important Proposition 8.1
asserts that rigid QPs have no oriented 2-cycles. Combining this with the fact that
rigidity is preserved by mutations, we conclude that every rigid QP is nondegenerate.
Using a result by Keller-Reiten [27], we show in Example 8.7 that the class of rigid
QPs (as well as the class of QPs with finite-dimensional Jacobian algebras) is strictly
greater than the class of QPs mutation-equivalent to acyclic ones. On the other hand,
QUIVERS WITH POTENTIALS I 5
Example 8.6 exhibits an underlying quiver without oriented 2-cycles that does not
admit a rigid QP; thus, the class of nondegenerate QPs is strictly greater than the
class of rigid ones.
In Section 9, we consider quivers that are mutation-equivalent to a Dynkin quiver.
For every such underlying quiver, we compute explicitly the corresponding rigid QP
(Proposition 9.1). Comparing this result with the description of cluster-tilted alge-
bras obtained in [13, 9], we conclude in Corollary 9.3 that in the case in question,
every cluster-tilted algebra can be identified with the Jacobian algebra of the cor-
responding rigid QP. Thus, Jacobian algebras can be viewed as generalizations of
cluster-tilted algebras.
In Section 10 we introduce decorated representations of QPs (QP-representations,
for short) and their right-equivalence (Definitions 10.1 and 10.2). We then present a
representation-theoretic extension of Splitting Theorem 4.6 by defining the reduced
part of a QP-representation M (Definition 10.4) and proving that, up to right-
equivalence, it is determined by the right-equivalence class of M (Proposition 10.5).
We use this result to introduce mutations of QP-representations and to prove a
representation-theoretic extension of Theorem 5.7: the mutation at every vertex is
an involution on the set of right-equivalence classes of reduced QP-representations
(Theorem 10.13).
Some examples of QP-representations and their mutations are given in Section 11.
All these examples treat quivers with three vertices. In particular, we describe the
effect of mutations on a special family of band representations coming from the theory
of string algebras [11, 21].
The concluding Section 12 contains some open problems about QPs and their
representations that we find essential for better understanding of the theory.
In the forthcoming continuation of this paper, we plan to discuss applications of
QP-representations and their mutations to the structure of the corresponding cluster
algebras.
2. Quivers and path algebras
A quiver Q = (Q0, Q1, h, t) consists of a pair of finite sets Q0 (vertices) and Q1
(arrows) supplied with two maps h : Q1 → Q0 (head) and t : Q1 → Q0 (tail). It
is represented as a directed graph with the set of vertices Q0 and directed edges
a : ta → ha for a ∈ Q1. Note that this definition allows the underlying graph to
have multiple edges and (multiple) loops.
We fix a field K, and associate to a quiver Q two vector spaces R = KQ0 and
A = KQ1 consisting of K-valued functions on Q0 and Q1, respectively. We will
sometimes refer to R as the vertex span of Q, and to A as the arrow span of Q. The
space R is a commutative algebra under the pointwise multiplication of functions.
The space A is an R-bimodule, with the bimodule structure defined as follows: if
e ∈ R and f ∈ A then (e ·f)(a) = e(ha)f(a) and (f ·e)(a) = f(a)e(ta) for all a ∈ Q1.
We denote by Q⋆ the dual or opposite quiver Q⋆ obtained by reversing the arrows
in Q (i.e., replacing Q = (Q0, Q1, h, t) with Q
⋆ = (Q0, Q1, t, h)). The corresponding
arrow span is naturally identified with the dual bimodule A⋆ (the dual vector space
of A with the standard R-bimodule structure).
6 HARM DERKSEN, JERZY WEYMAN, AND ANDREI ZELEVINSKY
For a given vertex set Q0 with the vertex span R, every finite-dimensional R-
bimodule B is the arrow span of some quiver on Q0. To see this, consider the
elements ei ∈ R for i ∈ Q0 given by ei(j) = δi,j (the Kronecker delta symbol).
They form a basis of idempotents of R, hence every R-bimodule B has a direct sum
decomposition
i,j∈Q0
Bi,j ,
where Bi,j = eiBej ⊆ B for every i, j ∈ Q0. If B is finite-dimensional, we can
identify the (finite) set of arrows Q1 with a K-basis in B which is the union of bases
in all components Bi,j; under this identification, every a ∈ Q1 ∩ Bi,j has h(a) = i
and t(a) = j.
It is convenient to represent an R-bimodule B by a matrix of vector spaces (Bi,j)
whose rows and columns are labeled by vertices. In this model, the left (resp. right)
action of an element c =
i ciei ∈ R is given by the left (resp. right) multiplication
by the diagonal matrix with diagonal entries ci. And the tensor product over R is
given by a usual matrix multiplication: if B =
i,j Bi,j and C =
i,j Ci,j, then
(B ⊗R C)i,j =
(Bi,k ⊗ Ck,j).
Returning to a quiver Q with the arrow span A, for each nonnegative integer d,
let Ad denote the R-bimodule
Ad = A⊗R A⊗R · · · ⊗R A︸ ︷︷ ︸
with the convention A0 = R.
Definition 2.1. The path algebra of Q is defined as the (graded) tensor algebra
R〈A〉 =
For each i, j ∈ Q0, the component R〈A〉i,j = eiR〈A〉ej is called the space of paths
from j to i.
As above, we identify the set of arrows Q1 with some basis of A consisting of
homogeneous elements, that is, each a ∈ Q1 belongs to some component Ai,j. Then
for every d ≥ 1, the products a1 · · · ad such that all ak belong to Q1, and t(ak) =
h(ak+1) for 1 ≤ k < d, form a K-basis of A
d. We call this basis the path basis of Ad
associated to Q1. For d = 0, we call {ei | i ∈ Q0} the path basis of A
0 = R. We
refer to the union of path bases for all d as the path basis of R〈A〉. The elements of
the path basis will be sometimes referred to simply as paths. We depict a1 · · · ad as
a path of length d starting in the vertex t(ad) and ending in h(a1). Note that the
product (a1 · · · ad)(ad+1 · · ·ad+k) of two paths is 0 unless t(ad) = h(ad+1), in which
case the product is given by concatenation of paths. This description implies the
following:
(2.1) If 0 6= p ∈ Akei and 0 6= q ∈ eiA
ℓ for some vertex i then pq 6= 0.
QUIVERS WITH POTENTIALS I 7
Definition 2.2. The complete path algebra of Q is defined as
R〈〈A〉〉 =
Thus, the elements of R〈〈A〉〉 are (possibly infinite) K-linear combinations of the
elements of a path basis in R〈A〉; and the multiplication in R〈〈A〉〉 naturally extends
the multiplication in R〈A〉.
Note that, if the quiver Q is acyclic (that is, has no oriented cycles), then Ad = {0}
for d≫ 0, hence in this case R〈〈A〉〉 = R〈A〉, and this algebra is finite-dimensional.
Example 2.3. Consider the quiver Q = (Q0, Q1) with Q0 = {1} and Q1 = {a} with
a : 1 → 1. This is the loop quiver:
1a 88 .
In this case R = KQ0 = K, and A = KQ1 = Ka. We have R〈A〉 = K[a], and
R〈〈A〉〉 = K[[a]], the algebra of formal power series.
Let m = m(A) denote the (two-sided) ideal of R〈〈A〉〉 given by
(2.2) m = m(A) =
Thus the powers of m are given by
We view R〈〈A〉〉 as a topologicalK-algebra via the m-adic topology having the powers
of m as a basic system of open neighborhoods of 0. Thus, the closure of any subset
U ⊆ R〈〈A〉〉 is given by
(2.3) U =
(U +mn).
It is clear that R〈A〉 is a dense subalgebra of R〈〈A〉〉.
In dealing with R〈〈A〉〉, the following fact is quite useful: every (non-commutative)
formal power series over R in a finite number of variables can be evaluated at arbi-
trary elements of m to obtain a well-defined element of R〈〈A〉〉. To illustrate, let us
show that m is the unique maximal two-sided ideal of R〈〈A〉〉 having zero intersec-
tion with R = A0. Indeed, it is enough to show that any element x ∈ R〈〈A〉〉 − m
generates an ideal having nonzero intersection with R. Let x = c + y with c a
nonzero element of R, and y ∈ m. Multiplying x on both sides by suitable elements
of R, we can assume that c = ei for some i ∈ Q0, and eiy = yei = y. But then
z = ei − y + y
2 − y3 + · · · is a well-defined element of R〈〈A〉〉, and we have xz = ei,
proving our claim.
This characterization of m implies that it is invariant under any algebra automor-
phism ϕ of R〈〈A〉〉 such that ϕ|R is the identity. Thus, ϕ is continuous, i.e., is an
automorphism of R〈〈A〉〉 as a topological algebra.
8 HARM DERKSEN, JERZY WEYMAN, AND ANDREI ZELEVINSKY
The same argument shows that, more generally, if A and A′ are finite-dimensional
R-bimodules then any algebra homomorphism ϕ : R〈〈A〉〉 → R〈〈A′〉〉 such that
ϕ|R = id, sends m(A) intom(A
′), hence is continuous. Thus, ϕ is uniquely determined
by its restriction to A1 = A, which is a R-bimodule homomorphism A → m(A′) =
A′ ⊕m(A′)2. We write ϕ|A = (ϕ
(1), ϕ(2)), where ϕ(1) : A→ A′ and ϕ(2) : A→ m(A′)2
are R-bimodule homomorphisms.
Proposition 2.4. Any pair (ϕ(1), ϕ(2)) of R-bimodule homomorphisms ϕ(1) : A→ A′
and ϕ(2) : A → m(A′)2 gives rise to a unique homomorphism of topological algebras
ϕ : R〈〈A〉〉 → R〈〈A′〉〉 such that ϕ|R = id, and ϕ|A = (ϕ
(1), ϕ(2)). Furthermore, ϕ is
an isomorphism if and only if ϕ(1) is a R-bimodule isomorphism A→ A′.
Proof. The uniqueness of ϕ is clear. To show the existence, let Q1 = {a1, . . . , aN} ⊂
A = A1 be the set of arrows in A. Writing an element x ∈ R〈〈A〉〉 as an infinite
K-linear combination of the elements of the corresponding path basis in R〈A〉, we
express x as a (non-commutative) formal power series F (a1, . . . , aN ). Therefore,
a desired algebra homomorphism can be obtained by setting ϕ(x) = F (ϕ(1)(a1) +
ϕ(2)(a1), . . . , ϕ
(1)(aN) + ϕ
(2)(aN)).
If ϕ is an isomorphism then ϕ(1) : A → A′ is clearly an isomorphism of R-
bimodules. To show the converse implication, we can identify A and A′ with the
help of ϕ(1), and so assume that ϕ(1) is the identity automorphism of A. Then the
(infinite) matrix that expresses ϕ as a K-linear map in the path basis of R〈〈A〉〉
is lower-triangular with all the diagonal entries equal to 1 (we order the basis ele-
ments so that their degrees weakly increase). Clearly, such a matrix is invertible,
completing the proof of Proposition 2.4. �
Definition 2.5. Let ϕ be the automorphism of R〈〈A〉〉 corresponding to a pair
(ϕ(1), ϕ(2)) as in Proposition 2.4. If ϕ(2) = 0, then we call ϕ a change of arrows. If ϕ(1)
is the identity automorphism of A, we say that ϕ is a unitriangular automorphism;
furthermore, we say that ϕ is of depth d ≥ 1, if ϕ(2)(A) ⊂ md+1.
The following property of unitriangular automorphisms is immediate from the
definitions:
If ϕ is an unitriangular automorphism of R〈〈A〉〉 of depth d,(2.4)
then ϕ(u)− u ∈ mn+d for u ∈ mn.
3. Potentials and their Jacobian ideals
In this section we introduce some of our main objects of study: potentials and
their Jacobian ideals in the complete path algebra R〈〈A〉〉 given by Definition 2.2.
We fix a path basis in R〈A〉; recall that it consists of the elements ei ∈ R = A
together with the products a1 · · · ad (paths) such that all ak belong to Q1, and t(ak) =
h(ak+1) for 1 ≤ k < d. Then each space A
d has a direct R-bimodule decomposition
i,j∈Q0
Adi,j, where the component A
i,j is spanned by the paths a1 · · · ad with
h(a1) = i and t(ad) = j.
QUIVERS WITH POTENTIALS I 9
Definition 3.1.
• For each d ≥ 1, we define the cyclic part of Ad as the sub-R-bimodule Adcyc =⊕
Adi,i. Thus, A
cyc is the span of all paths a1 · · · ad with h(a1) = t(ad);
we call such paths cyclic.
• We define a closed vector subspace R〈〈A〉〉cyc ⊆ R〈〈A〉〉 by setting
R〈〈A〉〉cyc =
Adcyc,
and call the elements of R〈〈A〉〉cyc potentials.
• For every ξ ∈ A⋆, we define the cyclic derivative ∂ξ as the continuous K-linear
map R〈〈A〉〉cyc → R〈〈A〉〉 acting on paths by
(3.1) ∂ξ(a1 · · · ad) =
ξ(ak)ak+1 · · · ada1 · · · ak−1.
• For every potential S, we define its Jacobian ideal J(S) as the closure of the
(two-sided) ideal in R〈〈A〉〉 generated by the elements ∂ξ(S) for all ξ ∈ A
(see (2.3)); clearly, J(S) is a two-sided ideal in R〈〈A〉〉.
• We call the quotient R〈〈A〉〉/J(S) the Jacobian algebra of S, and denote it
by P(Q, S) or P(A, S).
An easy check shows that a cyclic derivative ∂ξ : R〈〈A〉〉cyc → R〈〈A〉〉 does not de-
pend on the choice of a path basis. Furthermore, cyclic derivatives do not distinguish
between the potentials that are equivalent in the following sense.
Definition 3.2. Two potentials S and S ′ are cyclically equivalent if S−S ′ lies in the
closure of the span of all elements of the form a1 · · ·ad − a2 · · · ada1, where a1 · · · ad
is a cyclic path.
The following proposition is immediate from (3.1).
Proposition 3.3. If two potentials S and S ′ are cyclically equivalent, then ∂ξ(S) =
′) for all ξ ∈ A⋆, hence J(S) = J(S ′) and P(A, S) = P(A, S ′).
It is easy to see that the definition of cyclical equivalence does not depend on the
choice of a path basis. In fact, it can be given in more invariant terms as follows.
Definition 3.4. For any topological K-algebra U , its trace space Tr(U) is defined as
Tr(U) = U/{U, U}, where {U, U} is the closure of the vector subspace in U spanned
by all commutators. We denote by π = πU : U → Tr(U) the canonical projection.
The following proposition is a direct consequence of the definitions.
Proposition 3.5. Two potentials S and S ′ are cyclically equivalent if and only if
πR〈〈A〉〉(S) = πR〈〈A〉〉(S
′). Thus, the Jacobian ideal and the Jacobian algebra of a
potential S depend only on the image of S in Tr(R〈〈A〉〉).
Recall that we identify the set of arrows Q1 with a K-basis in A = A
1. For a ∈ Q1,
we will use the notation ∂a for the cyclic derivative ∂a⋆ , where Q
1 = {a
⋆ | a ∈ Q1} is
the dual basis of Q1 in A
10 HARM DERKSEN, JERZY WEYMAN, AND ANDREI ZELEVINSKY
Example 3.6. Consider the quiver Q = (Q0, Q1) with Q0 = {1, 2} and Q1 = {a, b},
where a : 1 → 2 and b : 2 → 1:
The vertex and arrow spans of Q are given by R = KQ0 = Ke1 ⊕ Ke2, and A =
KQ1 = Ka⊕Kb. The paths in R〈〈A〉〉 are e1, e2 and all products of the generators
a and b in which the factors a and b alternate. The potentials are (possibly infinite)
linear combinations of the elements (ab)n and (ba)n for all n ≥ 1. Using (3.1), we
obtain
∂a((ab)
n) = ∂a((ba)
n) = nb(ab)n−1, ∂b((ab)
n) = ∂b((ba)
n) = na(ba)n−1 (n ≥ 1).
Up to cyclical equivalence, every potential can be written in the form
αn(ab)
n (αn ∈ K).
Returning to the general theory, it is clear that every algebra homomorphism
ϕ : R〈〈A〉〉 → R〈〈A′〉〉
such that ϕ|R = id, sends potentials to potentials.
Proposition 3.7. Every algebra isomorphism
ϕ : R〈〈A〉〉 → R〈〈A′〉〉
such that ϕ|R = id, sends J(S) onto J(ϕ(S)), inducing an isomorphism of Jacobian
algebras P(A, S) → P(A′, ϕ(S)).
Proof. We start by developing some “differential calculus” for cyclic derivatives. We
need a few pieces of notation. We set
R〈〈A〉〉⊗̂R〈〈A〉〉 =
d,e≥0
(Ad ⊗ Ae)
(the tensor product on the right is over the base field K), and view this space as a
topological vector space with a basic system of open neighborhoods of 0 formed by
the sets
d+e≥n(A
d⊗Ae) for all n ≥ 0; thus, R〈A〉⊗R〈A〉 is dense inR〈〈A〉〉⊗̂R〈〈A〉〉.
Now, for every ξ ∈ A⋆, we define a continuous K-linear map
∆ξ : R〈〈A〉〉 → R〈〈A〉〉⊗̂R〈〈A〉〉
by setting ∆ξ(e) = 0 for e ∈ R = A
0, and
(3.2) ∆ξ(a1 · · · ad) =
ξ(ak)a1 · · · ak−1 ⊗ ak+1 · · · ad
for any path a1 · · · ad of length d ≥ 1. Note that ∆ξ does not depend on the choice of
a path basis. We will use the same convention as for cyclic derivatives: for a ∈ Q1,
we write ∆a instead of ∆a⋆ . For instance, in the situation of Example 3.6, we have
∆a((ab)
(ab)k−1 ⊗ b(ab)n−k, ∆b((ab)
(ab)k−1a⊗ (ab)n−k.
QUIVERS WITH POTENTIALS I 11
Next, we denote by (f, g) 7→ f�g a continuousK-bilinear map (R〈〈A〉〉⊗̂R〈〈A〉〉)×
R〈〈A〉〉 → R〈〈A〉〉 given by
(3.3) (u⊗ v)�g = vgu
for u, v ∈ R〈A〉. We are now ready to state the Leibniz rule.
Lemma 3.8 (Cyclic Leibniz rule). Let f ∈ R〈〈A〉〉i,j and g ∈ R〈〈A〉〉j,i for some
vertices i and j. Then for every ξ ∈ A⋆, we have
(3.4) ∂ξ(fg) = ∆ξ(f)�g +∆ξ(g)�f.
More generally, for any finite sequence of vertices i1, . . . , id, id+1 = i1 and for any
f1, . . . fd such that fk ∈ R〈〈A〉〉ik,ik+1, we have
(3.5) ∂ξ(f1 · · ·fd) =
∆ξ(fk)�(fk+1 · · · fdf1 · · · fk−1).
Proof. It is enough to check (3.4) in the case where f = a1 · · · ad and g = ad+1 · · · ad+s
are two paths such that t(ad) = h(ad+1) and t(ad+s) = h(a1). Using (3.1), we obtain
∂ξ(fg) =
ξ(ak)ak+1 · · · ad+sa1 · · · ak−1.
Comparing this expression with (3.2) and (3.3), we see that the part of the last sum
where k runs from 1 to d (resp. from d + 1 to d + s) is equal to ∆ξ(f)�g (resp. to
∆ξ(g)�f), proving (3.4). The identity (3.5) follows from (3.4) by induction on d. �
Lemma 3.9 (Cyclic chain rule). Suppose that ϕ : R〈〈A〉〉 → R〈〈A′〉〉 is an algebra
homomorphism as in Proposition 2.4. Then, for every potential S ∈ R〈〈A〉〉)cyc and
ξ ∈ A′⋆, we have:
(3.6) ∂ξ(ϕ(S)) =
∆ξ(ϕ(a))�ϕ(∂a(S)).
Proof. It suffices to treat the case where S = a1 · · ·ad is a cyclic path. Applying
(3.5) and (3.1), we obtain
∂ξ(ϕ(S)) =
∆ξ(ϕ(ak))�(ϕ(ak+1 · · · ada1 · · · ak−1))
∆ξ(ϕ(a))�ϕ(
k:ak=a
ak+1 · · ·ada1 · · · ak−1)
∆ξ(ϕ(a))�ϕ(∂a(S)),
as desired. �
Now we are ready to prove Proposition 3.7. By Lemma 3.9, for every ξ ∈ A′⋆, the
element ∂ξ(ϕ(S)) lies in the ideal generated by the elements ϕ(∂a(S)) for a ∈ Q1,
hence, it lies in ϕ(J(S)). Thus, we have the inclusion
J(ϕ(S)) ⊆ ϕ(J(S)).
12 HARM DERKSEN, JERZY WEYMAN, AND ANDREI ZELEVINSKY
We can also apply this to the inverse isomorphism ϕ−1 and the potential ϕ(S):
J(S) = J(ϕ−1(ϕ(S))) ⊆ ϕ−1(J(ϕ(S)).
Applying ϕ to both sides yields
ϕ(J(S)) ⊆ J(ϕ(S)),
completing the proof. �
4. Quivers with potentials
We now introduce our main objects of study.
Definition 4.1. Suppose Q is a quiver with the arrow span A, and S ∈ R〈〈A〉〉cyc
is a potential. We say that a pair (Q, S) (or (A, S)) is a quiver with potential (QP
for short) if it satisfies the following two conditions:
(4.1) The quiver Q has no loops, i.e., Ai,i = 0 for all i ∈ Q0.
(4.2) No two cyclically equivalent cyclic paths appear in the decomposition of S.
In view of (4.1), every potential S belongs to m(A)2; and condition (4.2) excludes,
for instance, any non-zero potential S cyclically equivalent to 0.
Definition 4.2. Let (A, S) and (A′, S ′) be QPs on the same vertex set Q0. By
a right-equivalence between (A, S) and (A′, S ′) we mean an algebra isomorphism
ϕ : R〈〈A〉〉 → R〈〈A′〉〉 such that ϕ|R = id, and ϕ(S) is cyclically equivalent to S
(see Definition 3.2).
In view of Proposition 3.5, any algebra homomorphism R〈〈A〉〉 → R〈〈A′〉〉 such
that ϕ|R = id, sends cyclically equivalent potentials to cyclically equivalent ones. It
follows that right-equivalences of QPs have the expected properties: the composition
of two right-equivalences, as well as the inverse of a right-equivalence, is again a
right-equivalence. Note also that an isomorphism ϕ : R〈〈A〉〉 → R〈〈A′〉〉 induces
an isomorphism of R-bimodules A and A′ (cf. Proposition 2.4), so in dealing with
right-equivalent QPs we can assume without loss of generality that A = A′.
In view of Propositions 3.3 and 3.7, any right-equivalence of QPs (A, S) ∼= (A′, S ′)
induces an isomorphism of the Jacobian ideals J(S) ∼= J(S ′) and of the Jacobian
algebras P(A, S) ∼= P(A′, S ′).
For every two QPs (A, S) and (A′, S ′) (on the same set of vertices Q0), we can
form their direct sum (A, S) ⊕ (A′, S ′) = (A ⊕ A′, S + S ′); it is well-defined since
both complete path algebras R〈〈A〉〉 and R〈〈A′〉〉 have canonical embeddings into
R〈〈A⊕ A′〉〉 as closed R-subalgebras.
We start our analysis of QPs with the case S ∈ A2. In this case, J(S) is the
closure of the ideal generated by the subspace
(4.3) ∂S = {∂ξ(S) | ξ ∈ A
⋆} ⊆ A.
Definition 4.3. We say that a QP (A, S) is trivial if S ∈ A2, and ∂S = A, or,
equivalently, P(A, S) = R.
The following description of trivial QPs is seen by standard linear algebra.
QUIVERS WITH POTENTIALS I 13
Proposition 4.4. A QP (A, S) with S ∈ A2 is trivial if and only if the set of arrows
Q1 consists of 2N distinct arrows a1, b1, . . . , aN , bN such that each akbk is a cyclic
2-path, and there is a change of arrows ϕ (see Definition 2.5) such that ϕ(S) is
cyclically equivalent to a1b1 + · · ·+ aNbN .
Returning to general QPs, we now show that taking direct sums with trivial ones
does not affect the Jacobian algebra.
Proposition 4.5. If (A, S) is an arbitrary QP, and (C, T ) is a trivial one, then
the canonical embedding R〈〈A〉〉 → R〈〈A⊕C〉〉 induces an isomorphism of Jacobian
algebras P(A, S) → P(A⊕ C, S + T ).
Proof. Let L denote the closure of the two-sided ideal in R〈〈A ⊕ C〉〉 generated by
C; thus, L is the set of all (possibly infinite) linear combinations of paths, each of
which contains at least one arrow from C. The definitions readily imply that
R〈〈A⊕ C〉〉 = R〈〈A〉〉 ⊕ L,
J(S + T ) = J(S)⊕ L
(in the last equality, J(S) is understood as the Jacobian ideal of S in R〈〈A〉〉).
Therefore,
P(A⊕ C, S + T ) = R〈〈A⊕ C〉〉/J(S + T ) = (R〈〈A〉〉 ⊕ L)/(J(S)⊕ L)
∼= R〈〈A〉〉/J(S) = P(A, S),
as desired. �
For an arbitrary QP (A, S), we denote by S(2) ∈ A2 the degree 2 homogeneous
component of S. We call (A, S) reduced if S(2) = 0, i.e., S ∈ m(A)3. We define
the trivial and reduced arrow spans of (A, S) as the finite-dimensional R-bimodules
given by
(4.4) Atriv = Atriv(S) = ∂S
(2), Ared = Ared(S) = A/∂S
(2) .
(see (4.3)).
The following statement will play a crucial role in later sections.
Theorem 4.6 (Splitting Theorem). For every QP (A, S) with the trivial arrow
span Atriv and the reduced arrow span Ared, there exist a trivial QP (Atriv, Striv)
and a reduced QP (Ared, Sred) such that (A, S) is right-equivalent to the direct sum
(Atriv, Striv)⊕(Ared, Sred). Furthermore, the right-equivalence class of each of the QPs
(Atriv, Striv) and (Ared, Sred) is determined by the right-equivalence class of (A, S).
Let us first prove the existence of a desired right-equivalence
(4.5) (A, S) ∼= (Atriv, Striv)⊕ (Ared, Sred).
There is nothing to prove if (A, S) is reduced, so let us assume that S(2) 6= 0. Using
Proposition 4.4 and replacing S if necessary by a cyclically equivalent potential, we
can assume that S is of the form
(4.6) S =
(akbk + akuk + vkbk) + S
14 HARM DERKSEN, JERZY WEYMAN, AND ANDREI ZELEVINSKY
where each akbk is a cyclic 2-path, the arrows a1, b1, . . . , aN , bN form a basis of Atriv,
the elements uk and vk belong tom
2, and the potential S ′ ∈ m3 is a linear combination
of cyclic paths containing none of the arrows ak or bk. The existence of a right-
equivalence (4.5) becomes a consequence of the following lemma.
Lemma 4.7. For every potential S of the form (4.6), there exists a unitriangular
automorphism ϕ of R〈〈A〉〉 such that ϕ(S) is cyclically equivalent to a potential of
the form (4.6) with uk = vk = 0 for all k.
We say that a potential S is d-split if it is of the form (4.6) with uk, vk ∈ m
d+1 for
all k. To prove Lemma 4.7, we first show the following.
Lemma 4.8. Suppose a potential S is d-split for some d ≥ 1. There exists a unitri-
angular automorphism ϕ of R〈〈A〉〉 having depth d and such that ϕ(S) is cyclically
equivalent to a 2d-split potential S̃ with ϕ(S)− S̃ ∈ m2d+2.
Proof. Let us write S in the form (4.6) with uk, vk ∈ m
d+1. Let ϕ be the unitriangular
automorphism of R〈〈A〉〉 acting on arrows as follows:
ϕ(ak) = ak − vk, ϕ(bk) = bk − uk, ϕ(c) = c (c ∈ Q1 − {a1, b1, . . . , aN , bN}).
Then ϕ is of depth d, so by (2.4), for each k, we have
ϕ(uk) = uk + u
k, ϕ(vk) = vk + v
k ∈ m
2d+1).
Therefore, we obtain
ϕ(S) =
((ak − vk)(bk − uk) + (ak − vk)(uk + u
k) + (vk + v
k)(bk − uk)) + S
(akbk + aku
k + v
kbk) + S1 + S
where
S1 = −
(vkuk + vku
k + v
kuk) ∈ m
2d+2.
In view of Definition 3.2, S1 is cyclically equivalent to a potential of the form∑
k(aku
k + v
kbk) + S
′′, where u′′k, v
k ∈ m
2d+1, and S ′′ is a linear combination of
cyclic paths containing none of the ak or bk. Furthermore, we have
S1 − S
k + v
kbk) ∈ m
2d+2.
We see that the desired potential S̃ can be chosen as
(akbk + ak(u
k + u
k) + (v
k + v
k)bk) + S
′ + S ′′,
completing the proof of Lemma 4.8. �
Proof of Lemma 4.7. Starting with a potential S of the form (4.6) and using repeat-
edly Lemma 4.8, we construct a sequence of potentials S1, S2, . . . , and a sequence of
unitriangular automorphisms ϕ1, ϕ2, . . . , with the following properties:
(1) S1 = S.
(2) Sd is 2
d−1-split.
(3) ϕd is of depth 2
QUIVERS WITH POTENTIALS I 15
(4) ϕd(Sd) is cyclically equivalent to Sd+1, and ϕd(Sd)− Sd+1 ∈ m
2d+2.
By property (3), setting
(4.7) ϕ = lim
ϕnϕn−1 · · ·ϕ1,
we obtain a well defined unitriangular automorphism ϕ of R〈〈A〉〉; indeed, in view of
(2.4), for any u ∈ R〈〈A〉〉, if we write ϕnϕn−1 · · ·ϕ1(u) as
d=0 u
n with u
n ∈ A
then each homogeneous component u
n stabilizes as n→ ∞.
We claim that this automorphism ϕ satisfies the required properties in Lemma 4.7.
To see this, for d ≥ 1, denote Cd = ϕd(Sd)− Sd+1. By (4), Cd ∈ {R〈〈A〉〉, R〈〈A〉〉} ∩
2d+2 (recall from Definition 3.4 that {R〈〈A〉〉, R〈〈A〉〉} denotes the closure of the
vector subspace in R〈〈A〉〉 spanned by all commutators). Using (1), it is easy to see
ϕnϕn−1 · · ·ϕ1(S) = Sn+1 +
ϕnϕn−1 · · ·ϕd+1(Cd)
for every n ≥ 1; passing to the limit as n→ ∞ yields
ϕ(S) = lim
Sn + ϕ(
(ϕd · · ·ϕ1)
−1(Cd))
(the convergence of the series on the right is clear since any automorphism of R〈〈A〉〉
preserves the powers of m). We conclude that ϕ(S) is cyclically equivalent to
limn→∞ Sn. In view of (2), the latter element is of the form (4.6) with uk = vk = 0
for all k. This completes the proofs of Lemma 4.7 and of the existence of a right-
equivalence (4.5). �
The above argument makes it clear that the right-equivalence class of (Atriv, Striv)
is determined by the right-equivalence class of (A, S) . To prove Theorem 4.6, it
remains to show that the same is true for (Ared, Sred). Changing notation a little bit,
we need to prove the following.
Proposition 4.9. Let (A, S) and (A, S ′) be reduced QPs, and (C, T ) a trivial QP.
If (A⊕C, S+T ) is right-equivalent to (A⊕C, S ′+T ) then (A, S) is right-equivalent
to (A, S ′).
We deduce Proposition 4.9 from the following result of independent interest.
Proposition 4.10. Let (A, S) and (A, S ′) be reduced QPs such that S ′−S ∈ J(S)2.
Then we have:
(1) J(S ′) = J(S).
(2) (A, S) is right-equivalent to (A, S ′). More precisely, there exists an algebra
automorphism ϕ of R〈〈A〉〉 such that ϕ(S) is cyclically equivalent to S ′, and
ϕ(u)− u ∈ J(S) for all u ∈ R〈〈A〉〉.
Proof. (1) Since (A, S) is reduced, we have J(S) ⊆ m2. As an easy consequence of
the cyclic Leibniz rule (3.4), we see that
∂ξ(J(S)
2)cyc ⊆ mJ(S) + J(S)m
16 HARM DERKSEN, JERZY WEYMAN, AND ANDREI ZELEVINSKY
for any ξ ∈ A⋆. It follows that
(4.8) ∂ξS
′ − ∂ξS ∈ mJ(S) + J(S)m,
implying that J(S ′) ⊆ J(S).
To show the reverse inclusion, note that (4.8) also implies that
J(S) ⊆ J(S ′) + (mJ(S) + J(S)m).
Applying the same inclusion to each of the terms J(S) on the right, we obtain
J(S) ⊆ J(S ′) + (m2J(S) +mJ(S)m+ J(S)m2).
Continuing in the same way, we get
J(S) ⊆ J(S ′) +
kJ(S)mn−k ⊆ J(S ′) +mn+2
for any n ≥ 1. Remembering the definition of topology in R〈〈A〉〉 (see (2.3)) and the
fact that J(S ′) is closed, we conclude that J(S) ⊆ J(S ′), finishing the proof of part
(1) of Proposition 4.10.
(2) Let Q1 = {a1, . . . , aN} be the set of arrows (that is, a basis of A). Then
a unitriangular automorphism ϕ of R〈〈A〉〉 is specified by a N -tuple of elements
b1, . . . , bN ∈ m
2 such that
(4.9) ϕ(ak) = ak + bk (k = 1, . . . , N).
Lemma 4.11. Let (A, S) be a reduced QP, and let ϕ be a unitriangular automor-
phism of R〈〈A〉〉 given by (4.9). Then the potential ϕ(S) − S −
k=1 bk∂akS is
cyclically equivalent to an element of mI2, where I is the closure of the ideal in
R〈〈A〉〉 generated by b1, . . . , bN .
Proof. First consider the case where S = ak1 · · · akd is a cyclic path of length d ≥ 3.
Then ϕ(S) = (ak1 +bk1) · · · (akd +bkd). Expanding this product, we see that the term
that contains no factors bkj is equal to S, while the sum of the terms that contain ex-
actly one factor bkj is easily seen to be cyclically equivalent to
k=1 bk∂akS (cf.(3.1)),
and the rest of the terms are cyclically equivalent to elements of
k=1m(m
d−1∩I)bk.
Writing a general potential S ∈ m3 as a linear combination of cyclic paths,we see
that ϕ(S)− S −
bk∂akS is cyclically equivalent to
ckbk, where each ck is
of the form
with c
kℓ ∈ m
d−1 ∩ I. Since I is closed, each ck is a well-defined element of mI,
implying the assertion of Lemma 4.11. �
We will also need one more lemma whose proof will be given in Section 13.
Lemma 4.12. Let I be a closed ideal of R〈〈A〉〉, and J be the closure of an ideal
generated by finitely many elements f1, f2, . . . , fN , which are bi-homogeneous with
respect to the vertex bigrading. Then every potential belonging to the ideal IJ is
cyclically equivalent to an element of the form
k=1 bkfk, where all bk belong to I.
QUIVERS WITH POTENTIALS I 17
To prove part (2) of Proposition 4.10, we construct a sequence of N -tuples
(b1n, . . . , bNn) (n ≥ 1)
of elements of m2 and the corresponding unitriangular automorphisms ϕn of R〈〈A〉〉
(so that ϕn(ak) = ak + bkn for k = 1, . . . , N) such that, for all n ≥ 1, we have
(1) bkn ∈ m
n+1 ∩ J(S) for k = 1, . . . , N .
(2) S ′ is cyclically equivalent to ϕ0ϕ1 · · ·ϕn−1(S +
k=1 bkn∂akS) (with the con-
vention that ϕ0 is the identity automorphism).
We proceed by induction on n. In the basic case n = 1, the existence of an
N -tuple (b11, . . . , bN1) with desired properties follows from Lemma 4.12 applied to
I = J = J(S) and fk = ∂akS (note that J(S) ⊆ m
2, since (A, S) is assumed to be
reduced).
Now assume that, for some n ≥ 1, we have already defined the elements bkℓ for
k = 1, . . . , N and ℓ = 1, . . . , n, satisfying (1) and (2). Applying Lemma 4.11 to
bk = bkn (so that ϕ = ϕn), we obtain that ϕn(S)− (S +
k=1 bkn∂akS) is cyclically
equivalent to an element of m(mn+1 ∩ J(S))2. We have
m(mn+1 ∩ J(S))2 ⊆ (mn+2 ∩ J(S))J(S).
This implies in particular that ϕn(S) − S is cyclically equivalent to an element of
J(S)2. Combining Proposition 3.7 with the already proved part (1) of Proposi-
tion 4.10, we conclude that ϕn(J(S)) = J(ϕn(S)) = J(S). It follows that ϕn(S) −
k=1 bkn∂akS) is cyclically equivalent to an element of ϕn((m
n+2 ∩ J(S))J(S)).
Applying Lemma 4.12 to I = mn+2 ∩ J(S), J = J(S) and fk = ∂akS, we see that
every potential in (mn+2 ∩ J(S))J(S) is cyclically equivalent to a potential of the
bk,n+1∂akS
for some bk,n+1 ∈ m
n+2∩J(S). It follows that S+
k=1 bkn∂akS is cyclically equivalent
to ϕn(S+
k=1 bk,n+1∂akS). Thus, conditions (1) and (2) get satisfied with n replaced
by n+ 1, completing our inductive step.
In view of condition (1), limn→∞ ϕ1 · · ·ϕn is a well-defined automorphism ϕ of
R〈〈A〉〉 such that ϕ(u)− u ∈ J(S) for all u ∈ R〈〈A〉〉. Passing to the limit n → ∞
in condition (2), we conclude that S ′ is cyclically equivalent to ϕ(S), completing the
proof of part (2) of Proposition 4.10. �
Proof of Proposition 4.9. We abbreviate J = J(S) and J ′ = J(S ′) (understood as
the Jacobian ideals of S and S ′ in R〈〈A〉〉). As in Proposition 4.5, let L denote the
closure of the two-sided ideal in R〈〈A⊕ C〉〉 generated by C. Then we have
(4.10) R〈〈A⊕ C〉〉 = R〈〈A〉〉 ⊕ L, J(S + T ) = J ⊕ L, J(S ′ + T ) = J ′ ⊕ L.
Let ϕ be an automorphism ofR〈〈A⊕C〉〉, such that ϕ(S+T ) is cyclically equivalent
to S ′ + T . In view of (4.10) and Proposition 3.7, we have
(4.11) ϕ(J ⊕ L) = J ′ ⊕ L.
18 HARM DERKSEN, JERZY WEYMAN, AND ANDREI ZELEVINSKY
Let ψ : R〈〈A〉〉 → R〈〈A〉〉 denote the restriction to R〈〈A〉〉 of the composition
pϕ, where p is the projection of R〈〈A ⊕ C〉〉 onto R〈〈A〉〉 along L. In view of
Proposition 4.10, it suffices to show the following:
ψ is an automorphism of R〈〈A〉〉 such that(4.12)
S ′ − ψ(S) is cyclically equivalent to an element of ψ(J2)
(indeed, assuming (4.12) and using Proposition 3.7, we see that ψ(J2) = J(ψ(S))2,
hence one can apply Proposition 4.10 to potentials S ′ and ψ(S)).
Clearly, ψ is an algebra homomorphism, so can be represented by a pair (ψ(1), ψ(2))
as in Proposition 2.4. To show that ψ is an automorphism of R〈〈A〉〉, it suffices to
show that ψ(1) is an R-bimodule automorphism of A. By the definition, if we write
the R-bimodule automorphism ϕ(1) of A⊕ C as a matrix
ϕAA ϕAC
ϕCA ϕCC
then ψ(1) = ϕAA. Since
ϕ(C) ⊂ ϕ(J ⊕ L) = J ′ ⊕ L ⊆ m(A)2 ⊕ L,
it follows that ϕAC = 0, implying that ψ
(1) = ϕAA is an R-bimodule automorphism
of A, and that ψ is an automorphism of R〈〈A〉〉.
Since S ′+T is cyclically equivalent to ϕ(S+T ), the same is true for the potentials
obtained from them by applying the projection p; it follows that S ′−ψ(S) is cyclically
equivalent to pϕ(T ). Since T ∈ C2, the claim that S ′ − ψ(S) is cyclically equivalent
to an element of ψ(J2) follows from the fact that pϕ(L) ⊆ ψ(J), or, equivalently,
that ϕ(L) ⊆ ϕ(J) + L. Applying the inverse automorphism ϕ−1 to both sides, it
suffices to show that L ⊆ J + ϕ−1(L). Using the obvious symmetry between J and
J ′, it is enough to show the inclusion L ⊆ J ′ + ϕ(L).
Let us abbreviate M = m(A⊕ C), and I = J ′ + ϕ(L). Since ϕ(J) ⊆ J ′ ⊕ L, and
J ⊆ m(A)2, it follows that ϕ(J) ⊆ J ′ ⊕ (L ∩M2) = J ′ +ML + LM . Therefore, we
L ⊆ J ′ + L = ϕ(J) + ϕ(L) ⊆ I +ML+ LM.
Substituting this upper bound for L into its right hand side, we deduce the inclusion
L ⊆ I +M2L+MLM + LM2.
Continuing in the same way, for every n > 0, we have the inclusion
L ⊆ I +
MkLMn−k ⊆ I +Mn+1.
In view of (2.3), it follows that L is contained in I, the closure of I in R〈〈A⊕ C〉〉.
However, it is easy to see that I = J ′ + ϕ(L) is closed in R〈〈A ⊕ C〉〉 (indeed, the
closedness of I is equivalent to that of ϕ−1(I) = ϕ−1(J ′)+L, and so, by symmetry, it
is enough to show that ϕ(J)+L is closed; but this is clear since ϕ(J)+L = p−1(ψ(J))
is the inverse image of the closed ideal ψ(J) of R〈〈A〉〉). This completes the proofs
of Proposition 4.9 and Theorem 4.6. �
QUIVERS WITH POTENTIALS I 19
Definition 4.13. We call the component (Ared, Sred) in the decomposition (4.5)
the reduced part of a QP (A, S) (by Theorem 4.6, it is determined by (A, S) up to
right-equivalence).
Definition 4.14. We call a quiver Q (as well as its arrow span A) 2-acyclic if it has
no oriented 2-cycles, i.e., satisfies the following condition:
(4.13) For every pair of vertices i 6= j, either Ai,j = {0} or Aj,i = {0}.
In the rest of this section we study the conditions on a QP (A, S) guaranteeing
that its reduced part is 2-acyclic. We need some preparation.
For a quiver Q with the arrow span A, let C = C(A) denote the set of cyclic
paths on A up to cyclical equivalence. Thus, C is either empty (if Q has no oriented
cycles at all), or countable. The space of potentials up to cyclical equivalence is
naturally identified with KC. We say that a K-valued function on KC is polynomial
if it depends on finitely many components of a potential S and can be expressed as a
polynomial in these components. For a nonzero polynomial function F , we denote by
U(F ) ⊂ KC the set of all potentials S such that F (S) 6= 0. By a regular function on
U(F ) we mean a ratio of two polynomial functions on KC such that the denominator
vanishes nowhere on U(F ); in particular, any function of the form G/F n, where G
is a polynomial, is regular on U(F ). If A′ is the arrow span of another quiver Q′, we
say that a map KC(A) → KC(A
′) is polynomial if its every component is a polynomial
function; similarly, a map U(F ) → KC(A
′) is regular if its every component is a
regular function on U(F ).
Now suppose that the arrow span A satisfies (4.1), and let {a1, b1, . . . , aN , bN} be
any maximal collection of distinct arrows in Q such that bkak is a cyclic 2-path for
k = 1, . . . , N . Then the quiver obtained from Q by removing this collection of arrows
is clearly 2-acyclic. To such a collection we associate a nonzero polynomial function
on KC(A) given by
(4.14) Db1,...,bNa1,...,aN (S) = det(xbqap)p,q=1,...,N ,
where xbqap is the sum of the coefficients of bqap and apbq in a potential S, with the
convention that xbqap = 0 unless bqap is a cyclic 2-path.
Proposition 4.15. The reduced part (Ared, Sred) of a QP (A, S) is 2-acyclic if and
only if Db1,...,bNa1,...,aN (S) 6= 0 for some collection of arrows as above. Furthermore, if A
the arrow span of the quiver obtained from Q by removing all arrows a1, b1, . . . , aN , bN ,
then there exists a regular map H : U(Db1,...,bNa1,...,aN ) → K
C(A′) such that, for any QP
(A, S) with S ∈ U(Db1,...,bNa1,...,aN ), the reduced part (Ared, Sred) is right-equivalent to
(A′, H(S)).
The proof of Proposition 4.15 follows by tracing the construction of (Ared, Sred)
given in the proof of Lemma 4.7. Note that we use the following convention. If A is
2-acyclic from the start then the only collection {a1, b1, . . . , aN , bN} as above is the
empty set; in this case, the function Db1,...,bNa1,...,aN is understood to be equal to 1, and H
is just the identity mapping.
20 HARM DERKSEN, JERZY WEYMAN, AND ANDREI ZELEVINSKY
5. Mutations of quivers with potentials
Let (A, S) be a QP. Suppose that a vertex k ∈ Q0 does not belong to an oriented
2-cycle. In other words, k satisfies the following condition:
(5.1) For every vertex i, either Ai,k or Ak,i is zero.
Replacing S if necessary with a cyclically equivalent potential, we can also assume
(5.2) No cyclic path occurring in the expansion of S starts (and ends) at k.
Under these conditions, we associate to (A, S) a QP µ̃k(A, S) = (Ã, S̃) on the same
set of vertices Q0. We define the homogeneous components Ãi,j as follows:
(5.3) Ãi,j =
(Aj,i)
⋆ if i = k or j = k;
Ai,j ⊕Ai,kAk,j otherwise;
here the product Ai,kAk,j is understood as a subspace of A
2 ⊆ R〈〈A〉〉. Thus, the
R-bimodule Ã is given by
(5.4) Ã = ekAek ⊕ AekA⊕ (ekA)
⋆ ⊕ (Aek)
where we use the notation
(5.5) ek = 1− ek =
i∈Q0−{k}
We associate to Q1 the set of arrows Q̃1 in the following way:
• Take all the arrows c ∈ Q1 not incident to k.
• For each incoming arrow a and outgoing arrow b at k, create a “composite”
arrow [ba] corresponding to the product ba ∈∈ AekA.
• Replace each incoming arrow a (resp. each outgoing arrow b) at k by the
corresponding arrow a⋆ (resp. b⋆) oriented in the opposite way.
More formally, for i = k or j = k, we set
(5.6) Q̃1 ∩ Ãi,j = {a
⋆ | a ∈ Q1 ∩Aj,i}
(the dual basis); and for i and j different from k, we define
(5.7) Q̃1 ∩ Ãi,j = (Q1 ∩ Ai,j)
{[ba] | b ∈ Q1 ∩ Ai,k, a ∈ Q1 ∩ Ak,j},
where [ba] ∈ Q̃1 ∩Ai,kAk,j denotes the arrow in Q̃1 associated with the product ba.
We now associate to S the potential µ̃k(S) = S̃ ∈ R〈〈Ã〉〉 given by
(5.8) S̃ = [S] + ∆k,
where
(5.9) ∆k = ∆k(A) =
a,b∈Q1: h(a)=t(b)=k
[ba]a⋆b⋆,
and [S] is obtained by substituting [apap+1] for each factor apap+1 with t(ap) =
h(ap+1) = k of any cyclic path a1 · · · ad occurring in the expansion of S (recall that
none of these cyclic paths starts at k). It is easy to see that both [S] and ∆k do not
depend on the choice of a basis Q1 of A.
QUIVERS WITH POTENTIALS I 21
The following proposition is immediate from the definitions.
Proposition 5.1. Suppose a QP (A, S) satisfies (5.1) and (5.2), and a QP (A′, S ′)
is such that ekA
′ = A′ek = {0}. Then we have
(5.10) µ̃k(A⊕ A
′, S + S ′) = µ̃k(A, S)⊕ (A
′, S ′).
Theorem 5.2. The right-equivalence class of the QP (Ã, S̃) = µ̃k(A, S) is deter-
mined by the right-equivalence class of (A, S).
Proof. Let Â be the finite-dimensional R-bimodule given by
(5.11) Â = A⊕ (ekA)
⋆ ⊕ (Aek)
The natural embedding A→ Â identifies R〈〈A〉〉 with a closed subalgebra in R〈〈Â〉〉.
We also have a natural embedding Ã → R〈〈Â〉〉 (sending each arrow [ba] to the
product ba). This allows us to identify R〈〈Ã〉〉 with another closed subalgebra in
R〈〈Â〉〉, namely, with the closure of the linear span of the paths â1 · · · âd such that
â1 /∈ ekA and âd /∈ Aek. Under this identification, the potential S̃ given by (5.8) and
viewed as an element of R〈〈Â〉〉 is cyclically equivalent to the potential
S + (
b∈Q1∩Aek
b⋆b)(
a∈Q1∩ekA
aa⋆).
Taking this into account, we see that Theorem 5.2 becomes a consequence of the
following lemma.
Lemma 5.3. Every automorphism ϕ of R〈〈A〉〉 can be extended to an automorphism
ϕ̂ of R〈〈Â〉〉 satisfying
(5.12) ϕ̂(R〈〈Ã〉〉) = R〈〈Ã〉〉,
(5.13) ϕ̂(
a∈Q1∩ekA
aa⋆) =
a∈Q1∩ekA
aa⋆, ϕ̂(
b∈Q1∩Aek
b⋆b) =
b∈Q1∩Aek
In order to extend ϕ to an automorphism ϕ̂ of R〈〈Â〉〉, we need only to define the
elements ϕ̂(a⋆) and ϕ̂(b⋆) for all arrows a ∈ Q1 ∩ ekA and b ∈ Q1 ∩ Aek.
We first deal with ϕ̂(a⋆). Let Q1 ∩ ekA = {a1, . . . , as}. In view of Proposition 2.4,
the action of ϕ on these arrows is given by
(5.14)
ϕ(a1) ϕ(a2) · · · ϕ(as)
a1 a2 · · · as
(C0 + C1),
where:
• C0 is an invertible s × s matrix with entries in K such that its (p, q)-entry
is 0 unless t(ap) = t(aq);
• C1 is a s× s matrix whose (p, q)-entry belongs to m(A)t(ap),t(aq).
Note that C0 +C1 is invertible, and its inverse is of the same form: indeed, we have
(C0 + C1)
−1 = (I + C−10 C1)
−1C−10 = (I +
(−1)n(C−10 C1)
n)C−10 .
22 HARM DERKSEN, JERZY WEYMAN, AND ANDREI ZELEVINSKY
Now we define the elements ϕ̂(a⋆p) by setting
ϕ̂(a⋆1)
ϕ̂(a⋆2)
ϕ̂(a⋆s)
 = (C0 + C1)
 .
It follows that
ϕ̂(a1) ϕ̂(a2) · · · ϕ̂(as)
ϕ̂(a⋆1)
ϕ̂(a⋆2)
ϕ̂(a⋆s)
a1 a2 · · · as
 =
For b ∈ Q1 ∩ Aek, we define ϕ̂(b
⋆) in a similar way. Namely, let Q1 ∩ Aek =
{b1, . . . , bt}. As above, the action of ϕ on these arrows is given by
(5.15)
ϕ(b1)
ϕ(b2)
ϕ(bt)
 = (D0 +D1)
 ,
where:
• D0 is an invertible t × t matrix with entries in K such that its (p, q)-entry
is 0 unless h(bp) = h(bq);
• D1 is a t× t matrix whose (p, q)-entry belongs to m(A)h(bp),h(bq).
As above, we see that D0+D1 is invertible, and its inverse is of the same form. Now
we define the elements ϕ̂(b⋆q) by setting
ϕ̂(b⋆1) ϕ̂(b
2) · · · ϕ̂(b
b⋆1 b
2 · · · b
(D0 +D1)
It follows that
b⋆qbq) =
ϕ̂(b1)
⋆ ϕ̂(b⋆2) · · · ϕ̂(b
ϕ̂(b1)
ϕ̂(b2)
ϕ̂(bt)
b⋆1 b
2 · · · b
 =
b⋆qbq.
The condition (5.13) is then clearly satisfied; the construction also makes clear that
the automorphism ϕ̂ of R〈〈Â〉〉 preserves the subalgebra R〈〈Ã〉〉. As a consequence
QUIVERS WITH POTENTIALS I 23
of Proposition 2.4, ϕ̂ restricts to an automorphism of R〈〈Ã〉〉, verifying (5.12) and
completing the proofs of Lemma 5.3 and Theorem 5.2. �
Note that even if a QP (A, S) is assumed to be reduced, the QP µ̃k(A, S) = (Ã, S̃)
is not necessarily reduced because the component [S](2) ∈ Ã2 may be non-zero.
Combining Theorems 4.6 and 5.2, we obtain the following corollary.
Corollary 5.4. Suppose a QP (A, S) satisfies (5.1) and (5.2), and let µ̃k(A, S) =
(Ã, S̃). Let (A, S) be a reduced QP such that
(5.16) (Ã, S̃) ∼= (Ãtriv, S̃
(2))⊕ (A, S)
(see (4.5)). Then the right-equivalence class of (A, S) is determined by the right-
equivalence class of (A, S).
Definition 5.5. In the situation of Corollary 5.4, we use the notation µk(A, S) =
(A, S) and call the correspondence (A, S) 7→ µk(A, S) the mutation at vertex k.
Note that if a QP (A, S) satisfies (5.1) then the same is true for µ̃k(A, S) and
for µk(A, S). Thus, the mutation µk is a well-defined transformation on the set
of right-equivalence classes of reduced QPs satisfying (5.1). (With some abuse of
notation, we sometimes denote a right-equivalence class by the same symbol as any
of its representatives.)
Example 5.6. Consider the quiver Q with vertices {1, 2, 3, 4} and arrows a : 1 → 2,
b : 2 → 3, c : 3 → 4 and d : 4 → 1:
Let S = dcba. Let us perform the mutation at vertex 2. The arrow a is replaced by
e := a⋆ : 2 → 1, and b is replaced by f := b⋆ : 3 → 2. We also have a new arrow
g := [ba] : 1 → 3. So µ̃2(A) corresponds to the quiver with vertices {1, 2, 3, 4} and
arrows c, d, e, f, g:
g=[ba]
The potential µ̃2(S) = S̃ is given by
S̃ = dcg + gef ;
thus, µ̃2(A, S) is reduced, and we have µ̃2(A, S) = µ2(A, S).
Note that S̃ does not satisfy condition (5.2) with respect to vertex k = 3 since the
path gef starts and ends at 3. But we can fix this condition by replacing S̃ with
a cyclically equivalent potential, say S ′ = dcg + efg. Now let us mutate (Ã, S ′) at
24 HARM DERKSEN, JERZY WEYMAN, AND ANDREI ZELEVINSKY
vertex 3. The arrows c, f, g are replaced by c⋆ : 4 → 3, f ⋆ : 2 → 3 and g⋆ : 3 → 1,
respectively. We also add new arrows [cg] : 1 → 4 and [fg] : 1 → 2. Thus, µ̃3(Ã, S
has arrows {d, e, c⋆, f ⋆, g⋆, [cg], [fg]}:
c⋆ //
The potential µ̃3(S
′) is given by
′) = d[cg] + e[fg] + [fg]g⋆f ⋆ + [cg]g⋆c⋆.
It is not reduced, so to obtain the reduced QP µ3(Ã, S
′), we need to remove the
trivial part of µ̃3(Ã, S
′). The resulting quiver is as follows:
c⋆ // 3
Since it is acyclic (that is, has no oriented cycles), the corresponding potential is 0.
Our next result is that every mutation is an involution.
Theorem 5.7. The correspondence µk : (A, S) → (A, S) acts as an involution on
the set of right-equivalence classes of reduced QPs satisfying (5.1), that is, µ2k(A, S)
is right-equivalent to (A, S).
Proof. Let (A, S) be a reduced QP satisfying (5.1) and (5.2). Let µ̃k(A, S) = (Ã, S̃)
and µ̃2k(A, S) = µ̃k(Ã, S̃) = (
S). In view of Theorem 4.6 and Proposition 5.1, it is
enough to show that
(5.17) (
S) is right-equivalent to (A, S)⊕ (C, T ), where (C, T ) is a trivial QP.
Using (5.4) twice, and identifying (ekA)
⋆ with A⋆ek, and (Aek)
⋆ with ekA
⋆, where
A⋆ is the dual R-bimodule of A, we conclude that
(5.18)
A = A⊕ AekA⊕A
Furthermore, the basis of arrows in
A consists of the original set of arrows Q1 in A
together with the arrows [ba] ∈ AekA and [a
⋆b⋆] ∈ A⋆ekA
⋆ for a ∈ Q1 ∩ ekA and
b ∈ Q1 ∩Aek.
Remembering (5.8) and (5.9), we see that the potential
S is given by
(5.19)
S = [[S]] + [∆k(A)] + ∆k(Ã) = [S] +
a,b∈Q1: h(a)=t(b)=k
([ba][a⋆b⋆] + [a⋆b⋆]ba),
QUIVERS WITH POTENTIALS I 25
hence it is cyclically equivalent to
(5.20) S1 = [S] +
a,b∈Q1: h(a)=t(b)=k
([ba] + ba)[a⋆b⋆]
(recall that [S] is obtained by substituting [apap+1] for each factor apap+1 with t(ap) =
h(ap+1) = k of any cyclic path a1 · · · ad occurring in the path expansion of S). Let
us abbreviate
(C, T ) = (AekA⊕ A
a,b∈Q1: h(a)=t(b)=k
[ba][a⋆b⋆]).
This is a trivial QP (cf. Proposition 4.4); therefore to prove Theorem 5.7 it suffices
to show that the QP (
A, S1) given by (5.18) and (5.20) is right-equivalent to (A, S)⊕
(C, T ). We proceed in several steps.
Step 1: Let ϕ1 be the change of arrows automorphism of R〈〈
A〉〉 (see Defini-
tion 2.5) multiplying each arrow b ∈ Q1 ∩ Aek by −1, and fixing the rest of the
arrows in
A. Then the potential S2 = ϕ1(S1) is given by
S2 = [S] +
a,b∈Q1: h(a)=t(b)=k
([ba]− ba)[a⋆b⋆].
Step 2: Let ϕ2 be the unitriangular automorphism of R〈〈
A〉〉 (see Definition 2.5)
sending each arrow [ba] ∈ AekA to [ba] + ba, and fixing the rest of the arrows in
A. Remembering the definition of [S], it is easy to see that the potential ϕ2(S2) is
cyclically equivalent to a potential of the form
S3 = S +
a,b∈Q1: h(a)=t(b)=k
[ba]([a⋆b⋆] + f(a, b))
for some elements f(a, b) ∈ m(A⊕ AekA)
Step 3: Let ϕ3 be the unitriangular automorphism of R〈〈
A〉〉 sending each arrow
[a⋆b⋆] ∈ A⋆ekA
⋆ to [a⋆b⋆] − f(a, b), and fixing the rest of the arrows in
A. Then we
have ϕ3(S3) = S + T .
Combining these three steps, we conclude that the QP (
A, S1) is right-equivalent
A, S + T ) = (A, S)⊕ (C, T ), finishing the proof of Theorem 5.7. �
6. Some mutation invariants
In this section we fix a vertex k and study the effect of the mutation µk on the
Jacobian algebra P(A, S). We will use the following notation: for an R-bimodule B,
denote
(6.1) Bk̂,k̂ = ekBek =
i,j 6=k
(see (5.5)). Note that if B is a (topological) algebra then B
k̂,k̂
is a (closed) subalgebra
of B.
26 HARM DERKSEN, JERZY WEYMAN, AND ANDREI ZELEVINSKY
Proposition 6.1. Suppose a QP (A, S) satisfies (5.1) and (5.2), and let (Ã, S̃) =
µ̃k(A, S) be given by (5.4) and (5.8). Then the algebras P(A, S)k̂,k̂ and P(Ã, S̃)k̂,k̂
are isomorphic to each other.
Proof. In view of (5.4), we have
(6.2) Ãk̂,k̂ = Ak̂,k̂ ⊕ AekA.
Thus, the algebra R〈〈Ãk̂,k̂〉〉 is generated by the arrows c ∈ Q1 ∩ Ak̂,k̂ and [ba] for
a ∈ Q1∩ekA and b ∈ Q1∩Aek. The following fact is immediate from the definitions.
Lemma 6.2. The correspondence sending each c ∈ Q1 ∩ Ak̂,k̂ to itself, and each
generator [ba] to ba extends to an algebra isomorphism
R〈〈Ã
k̂,k̂
〉〉 → R〈〈A〉〉
k̂,k̂
Let u 7→ [u] denote the isomorphism R〈〈A〉〉k̂,k̂ → R〈〈Ãk̂,k̂〉〉 inverse of that in
Lemma 6.2. It acts in the same way as the correspondence S 7→ [S] in (5.8): [u] is
obtained by substituting [apap+1] for each factor apap+1 with t(ap) = h(ap+1) = k of
any path a1 · · ·ad occurring in the path expansion of u.
Lemma 6.3. The correspondence u 7→ [u] induces an algebra epimorphism
P(A, S)
k̂,k̂
→ P(Ã, S̃)
k̂,k̂
Proof. It is enough to prove the following two facts:
(6.3) R〈〈Ã〉〉k̂,k̂ = R〈〈Ãk̂,k̂〉〉+ J(S̃)k̂,k̂;
(6.4) [J(S)k̂,k̂] ⊆ R〈〈Ãk̂,k̂〉〉 ∩ J(S̃)k̂,k̂.
To show (6.3), we note that if a path ã1 · · · ãd ∈ R〈〈Ã〉〉k̂,k̂ does not belong to
R〈〈Ã
k̂,k̂
〉〉 then it must contain one or more factors of the form a⋆b⋆ with h(a) =
t(b) = k. In view of (5.8) and (5.9), we have
(6.5) a⋆b⋆ = ∂[ba]S̃ − ∂[ba][S].
Substituting this expression for each factor a⋆b⋆, we see that ã1 · · · ãd ∈ R〈〈Ãk̂,k̂〉〉+
J(S̃)k̂,k̂, as desired.
To show (6.4), we note that J(S)k̂,k̂ is easily seen to be the closure of the ideal
in R〈〈A〉〉
k̂,k̂
generated by the elements ∂cS for all arrows c ∈ Q1 with t(c) 6= k and
h(c) 6= k, together with the elements (∂aS)a
′ for a, a′ ∈ Q1 ∩ ekA , and b
′(∂bS) for
b, b′ ∈ Q1 ∩ Aek. Let us apply the map u 7→ [u] to these generators. First, we have:
(6.6) [∂cS] = ∂cS̃.
QUIVERS WITH POTENTIALS I 27
With a little bit more work (using (6.5)), we obtain
[(∂aS)a
t(b)=k
(∂[ba][S])[ba
t(b)=k
(∂[ba]S̃ − a
⋆b⋆)[ba′](6.7)
t(b)=k
(∂[ba]S̃)[ba
′]− a⋆∂a′⋆S̃,
[b′(∂bS)] =
h(a)=k
[b′a](∂[ba][S])
h(a)=k
[b′a](∂[ba]S̃ − a
⋆b⋆)(6.8)
h(a)=k
[b′a](∂[ba]S̃)− (∂b′⋆S̃)b
This implies the desired inclusion in (6.4). �
To finish the proof of Proposition 6.1, it is enough to show that the epimorphism
in Lemma 6.3 (let us denote it by α) is in fact an isomorphism. To do this, we
construct the left inverse algebra homomorphism β : P(Ã, S̃)
k̂,k̂
→ P(A, S)
k̂,k̂
that βα is the identity map on P(A, S)k̂,k̂). We define β as the composition of three
maps. First, we apply the epimorphism P(Ã, S̃)
k̂,k̂
k̂,k̂
defined in the same
way as α. Remembering the proof of Theorem 5.7 and using the notation introduced
there, we then apply the isomorphism P(
S)k̂,k̂ → P(A ⊕ C, S + T )k̂,k̂ induced
by the automorphism ϕ3ϕ2ϕ1 of R〈〈A ⊕ C〉〉. Finally, we apply the isomorphism
P(A⊕ C, S + T )k̂,k̂ → P(A, S)k̂,k̂ given in Proposition 4.5.
Since all the maps involved are algebra homomorphisms, it is enough to check
that βα fixes the generators p(c) and p(ba) of P(A, S)k̂,k̂, where p is the projection
R〈〈A〉〉 → P(A, S), and a, b, c have the same meaning as above. This is done by
direct tracing of the definitions. �
Proposition 6.4. In the situation of Proposition 6.1, if the Jacobian algebra P(A, S)
is finite-dimensional then so is P(Ã, S̃).
Proof. We start by showing that finite dimensionality of P(A, S) follows from a
seemingly weaker condition.
Lemma 6.5. Let J ⊆ m(A) be a closed ideal in R〈〈A〉〉. Then the quotient alge-
bra R〈〈A〉〉/J is finite dimensional provided the subalgebra R〈〈A〉〉k̂,k̂/Jk̂,k̂ is finite
dimensional. In particular, the Jacobian algebra P(A, S) is finite-dimensional if and
only if so is the subalgebra P(A, S)
k̂,k̂
Proof. Similarly to (6.1), for an R-bimodule B, we denote
Bk,k̂ = ekBek =
j 6=k
Bk,j, Bk̂,k = ekBek =
i 6=k
Bi,k.
28 HARM DERKSEN, JERZY WEYMAN, AND ANDREI ZELEVINSKY
We need to show that if R〈〈A〉〉k̂,k̂/Jk̂,k̂ is finite dimensional then so is each of the
spaces R〈〈A〉〉
, R〈〈A〉〉
andR〈〈A〉〉k,k/Jk,k. Let us treatR〈〈A〉〉k,k/Jk,k;
the other two cases are done similarly (and a little simpler).
Q1 ∩Ak,k̂ = {a1, . . . , as}, Q1 ∩ Ak̂,k = {b1, . . . , bt}.
We have
R〈〈A〉〉k,k = Kek ⊕
aℓR〈〈A〉〉k̂,k̂bm.
It follows that there is a surjective map α : K×Mats×t(R〈〈A〉〉k̂,k̂) → R〈〈A〉〉k,k/Jk,k
given by
α(c, C) = p(cek +
a1 a2 · · · as
),
where Mats×t(B) stands for the space of s× t matrices with entries in B, and p is the
projection R〈〈A〉〉 → R〈〈A〉〉/J . The kernel of α contains the space Mats×t(Jk̂,k̂),
hence R〈〈A〉〉k,k/Jk,k is isomorphic to a quotient of the finite-dimensional space K ×
Mats×t(R〈〈A〉〉k̂,k̂/Jk̂,k̂). Thus, R〈〈A〉〉k,k/Jk,k is finite dimensional, as desired. �
To finish the proof of Proposition 6.4, suppose that P(A, S) is finite dimensional.
Then P(Ã, S̃)k̂,k̂ is finite dimensional by Proposition 6.1. Applying Lemma 6.5 to
the QP (Ã, S̃), we conclude that P(Ã, S̃) is finite dimensional, as desired. �
Remembering (5.16) and using Proposition 4.5, we see that Propositions 6.1 and
6.4 have the following corollary.
Corollary 6.6. Suppose (A, S) is a reduced QP satisfying (5.1), and let (A, S) =
µk(A, S) be a reduced QP obtained from (A, S) by the mutation at k. Then the
algebras P(A, S)
k̂,k̂
and P(A, S)
k̂,k̂
are isomorphic to each other, and P(A, S) is
finite-dimensional if and only if so is P(A, S).
We see that the class of QPs with finite dimensional Jacobian algebras is invariant
under mutations. Let us now present another such class.
Definition 6.7. For every QP (A, S), we define its deformation space Def(A, S) by
(6.9) Def(A, S) = Tr(P(A, S))/R
(see Definitions 3.1 and 3.4).
Remark 6.8. Definition 6.7 can be motivated as follows (we keep the following
arguments informal although with some work they can be made rigorous). Let G =
Aut(R〈〈A〉〉) be the group of algebra automorphisms ofR〈〈A〉〉 (acting as the identity
on R). Using Proposition 2.4, we can think of G as an infinite dimensional algebraic
group. In view of Definition 3.4, G acts naturally on the trace space Tr(R〈〈A〉〉).
Remembering Definition 4.2, it is natural to think of the deformation space of a QP
(A, S) as the normal space at π(S) of the orbit G·π(S) in the ambient space π(m(A)2)
(recall that π stands for the natural projection R〈〈A〉〉 → Tr(R〈〈A〉〉)). Arguing as
QUIVERS WITH POTENTIALS I 29
in Lemma 4.11, we conclude that the infinitesimal action of (the Lie algebra of) G
on π(m(A)2) is by the transformations
π(u) 7→ π(
bk∂aku),
where Q1 = {a1, . . . , aN} is the set of arrows, and bk ∈ m(A)h(ak),t(ak) (this is well
defined in view of Proposition 3.3). This makes it natural to identify the tangent
space at π(S) of G · π(S) with π(J(S)), hence to identify the corresponding normal
space with π(m(A))/π(J(S)), or equivalently, with the space Def(S) given by (6.9).
Proposition 6.9. In the situation of Proposition 6.1, deformation spaces Def(Ã, S̃)
and Def(A, S) are isomorphic to each other.
Proof. In view of Proposition 3.5, Def(A, S) is isomorphic to Tr(P(A, S)k̂,k̂)/Rk̂,k̂.
Therefore, our assertion is immediate from Proposition 6.1. �
Definition 6.10. We call a QP (A, S) rigid if Def(A, S) = {0}, i.e., if Tr(P(A, S)) =
Combining Propositions 4.5 and 6.9, we obtain the following corollary.
Corollary 6.11. If a reduced QP (A, S) satisfies (5.1) and is rigid, then the QP
(A, S) = µk(A, S) is also rigid.
Some examples of rigid and non-rigid QPs will be given in Section 8.
7. Nondegenerate QPs
If we wish to be able to apply to a reduced QP (A, S) the mutation at every vertex
of Q0, the R-bimodule A must satisfy (5.1) at all vertices. Thus, the arrow span A
must be 2-acyclic (see Definition 4.14). Such an arrow span A can be encoded by a
skew-symmetric integer matrix B = B(A) = (bi,j) with rows and columns labeled by
Q0, by setting
(7.1) bi,j = dimAi,j − dimAj,i.
Indeed, the dimensions of the components Ai,j are recovered from B by
(7.2) dimAi,j = [bi,j ]+,
where we use the notation
(7.3) [x]+ = max(x, 0).
Proposition 7.1. Let (A, S) be a 2-acyclic reduced QP, and suppose that the reduced
QP µk(A, S) = (A, S) obtained from (A, S) by the mutation at some vertex k (see
Definition 5.5) is also 2-acyclic. Let B(A) = (bi,j) and B(A) = (bi,j) be the skew-
symmetric integer matrices given by (7.1). Then we have
(7.4) bi,j =
−bi,j if i = k or j = k;
bi,j + [bi,k]+ [bk,j ]+ − [−bi,k]+ [−bk,j ]+ otherwise.
30 HARM DERKSEN, JERZY WEYMAN, AND ANDREI ZELEVINSKY
Proof. First we note that by Proposition 4.4, if (C, T ) is a trivial QP then dimCi,j =
dimCj,i for all i, j. In view of (5.16), this implies that
(7.5) bi,j = dimAi,j − dimAj,i = dim Ãi,j − dim Ãj,i,
where (Ã, S̃) = µ̃k(A, S). Using (5.3), we obtain
dim Ãi,j =
dimAj,i if i = k or j = k;
dimAi,j + dimAi,k dimAk,j otherwise.
To obtain (7.4), it remains to substitute these expressions into (7.5) and use (7.2). �
An easy calculation using the obvious identity x = [x]+ − [−x]+ shows that the
second case in (7.4) can be rewritten in several equivalent ways as follows:
bi,j = bi,j + sgn(bi,k) [bi,kbk,j ]+
= bi,j + [−bi,k]+ bk,j + bi,k[bk,j]+
= bi,j +
|bi,k|bk,j + bi,k|bk,j|
It follows that the transformation B 7→ B given by (7.4) coincides with the matrix
mutation at k which plays a crucial part in the theory of cluster algebras (cf. [18,
(4.3)], [20, (2.2), (2.5)]).
We see that the mutations of 2-acyclic QPs provide a natural framework for matrix
mutations. With some abuse of notation, we denote by µk(A) the 2-acyclic R-
bimodule such that the skew-symmetric matrix B(µk(A)) is obtained from B(A) by
the mutation at k; thus, µk(A) is determined by A up to an isomorphism.
Note that the matrix mutations at arbitrary vertices can be iterated indefinitely,
while the 2-acyclicity condition (4.13) can be destroyed by a QP mutation. We will
study QPs for which this does not happen.
Definition 7.2. Let k1, . . . , kℓ ∈ Q0 be a finite sequence of vertices such that kp 6=
kp+1 for p = 1, . . . , ℓ − 1. We say that a QP (A, S) is (kℓ, · · · , k1)-nondegenerate if
all the QPs (A, S), µk1(A, S), µk2µk1(A, S), . . . , µkℓ · · ·µk1(A, S) are 2-acyclic (hence
well-defined). We say that (A, S) is nondegenerate if it is (kℓ, . . . , k1)-nondegenerate
for every sequence of vertices as above.
To state our next result recall the terminology introduced before Proposition 4.15.
In particular, for a given quiver with the arrow span A, the QPs on A are identified
with the elements of KC(A).
Proposition 7.3. Suppose that the base field K is infinite, Q is a 2-acyclic quiver
with the arrow span A, a sequence of vertices k1, . . . , kℓ is as in Definition 7.2, and
A′ = µkℓ · · ·µk1(A). Then there exist a non-zero polynomial function F : K
C(A) →
K and a regular map G : U(F ) → KC(A
′) such that every QP (A, S) with S ∈
U(F ) is (kℓ, . . . , k1)-nondegenerate, and, for any QP (A, S) with S ∈ U(F ), the QP
µkℓ · · ·µk1(A, S) is right-equivalent to (A
′, G(S)).
Proof. We proceed by induction on ℓ. First let us deal with the case ℓ = 1, that
is, with a single mutation µk. Recall that µk(A, S) = (A, S) is the reduced part of
the QP µ̃k(A, S) = (Ã, S̃) given by (5.3) and (5.8). It is clear from the definition
QUIVERS WITH POTENTIALS I 31
that S̃ = G̃(S) for a polynomial map G̃ : KC(A) → KC(
eA). Now let us apply
Proposition 4.15 to the quiver with the arrow span Ã. We see that there exists
a polynomial function of the form Dd1,...,dNc1,...,cN on K
C( eA) (see (4.14), where we have
changed the notation for the arrows to avoid the notation conflict with Section 5) such
that the reduced part (A, S) of a QP (Ã, S̃) is 2-acyclic whenever S̃ ∈ U(Dd1,...,dNc1,...,cN ).
Furthermore, for S̃ ∈ U(Dd1,...,dNc1,...,cN ), the QP (A, S) is right-equivalent to (A
′, H(S̃))
for some regular map H : U(Dd1,...,dNc1,...,cN ) → K
C(A′), where A′ = µk(A). We now define
a polynomial function F : KC(A) → K and a regular map G : U(F ) → KC(A
′) by
setting
(7.6) F = Dd1,...,dNc1,...,cN ◦ G̃, G = H ◦ G̃.
To finish the argument for ℓ = 1, it remains to show that F is not identically equal
to zero. But this is clear from the definitions (4.14) and (5.8), since the oriented 2-
cycles in Ã (up to cyclical equivalence) are of the form c[ba] and so are in a bijection
with the oriented 3-cycles cba in A that pass through k.
Now assume that ℓ ≥ 2, and that our assertion holds if we replace ℓ by ℓ− 1. Let
A1 = µk1(A), so A
′ = µkℓ · · ·µk2(A1). By the inductive assumption, there exist a non-
zero polynomial function F ′ : KC(A1) → K and a regular map G′ : U(F ′) → KC(A
such that, for any QP (A1, S1) with S1 ∈ U(F
′), the QP µkℓ · · ·µk2(A1, S1) is right-
equivalent to (A′, G′(S1)). Also by the already established case ℓ = 1, there exists a
non-zero polynomial function F ′′ : KC(A1) → K such that, for any QP (A1, S1) with
S1 ∈ U(F
′′), the QP µk1(A1, S1) is 2-acyclic, hence is right-equivalent to some QP
on A. Since the base field K is assumed to be infinite, we have U(F ′) ∩ U(F ′′) 6= ∅.
Choose S
1 ∈ U(F
′) ∩ U(F ′′), and let (A, S0) = µk(A1, S
1 ). By Theorem 5.7, we
have µk(A, S0) = (A1, S
1 ). By the above argument for ℓ = 1, there exist a nonzero
polynomial function F1 : K
C(A) → K and a regular map G1 : U(F1) → K
C(A1)
(of the type (7.6)) such that µk(A, S) = (A1, G1(S)) for S ∈ U(F1). In particular,
we have G1(S0) = S
1 implying that F
′ ◦ G1 is a nonzero polynomial function on
KC(A). It follows that the nonzero polynomial function F (S) = F1(S)F
′(G1(S)) and
the regular map G = G′ ◦ G1 : U(F ) → K
C(A′) are well-defined and satisfy all the
required conditions. This completes the proof of Proposition 7.3. �
Corollary 7.4. For every 2-acyclic arrow span A, there exists a countable family F
of nonzero polynomial functions on KC(A) such that the QP (A, S) is nondegenerate
whenever S ∈
F∈F U(F ). In particular, if the base field K is uncountable, then
there exists a nondegenerate QP on A.
Proof. By Proposition 7.3, for every sequence k1, . . . , kℓ as in Definition 7.2, there
exists a nonzero polynomial function Fk1,...,kℓ on K
C(A) such that a QP (A, S) is
(kℓ, . . . , k1)-nondegenerate for S ∈ U(Fk1,...,kℓ). These functions form a desired count-
able family F .
It remains to prove that
F∈F U(F ) 6= ∅ provided K is uncountable. If A is
acyclic, i.e., has no oriented cycles, then KC(A) = {0}, and each of the functions in
F is just a nonzero constant, so there is nothing to prove; no assumption on K is
needed here. If A has at least one oriented cycle then the set C(A) is countable (recall
32 HARM DERKSEN, JERZY WEYMAN, AND ANDREI ZELEVINSKY
that it consists of cyclic paths up to cyclical equivalence). Thus, we can realize KC(A)
as the polynomial ring K[X1, X2, . . . ] in countably many indeterminates. Since K is
uncountable, we can choose x1 so that F (x1) 6= 0 for all F ∈ F ∩K[X1]. Then we
choose x2 so that F (x1, x2) 6= 0 for all F ∈ F ∩K[X1, X2]. Continuing like this, we
find a sequence x1, x2, . . . such that F (x1, x2, . . . ) 6= 0 for all F ∈ F . �
8. Rigid QPs
Proposition 8.1. Every rigid reduced QP (A, S) is 2-acyclic.
Proof. First note that the definition of rigidity can be conveniently restated as fol-
lows:
a QP (A, S) is rigid if and only if every potential(8.1)
on A is cyclically equivalent to an element of J(S).
Now suppose for the sake of contradiction that for some i 6= j both components Ai,j
and Aj,i are non-zero. Choose non-zero elements a ∈ Ai,j and b ∈ Aj,i. Remembering
the definition of the Jacobian ideal (see Definition 3.1), it is easy to see that the cyclic
part of J(S) is contained in m(A)3. It follows that ab is not cyclically equivalent to
an element of J(S), in contradiction with (8.1). �
Combining Proposition 8.1 with Corollary 6.11, we obtain the following result.
Corollary 8.2. Any rigid QP is nondegenerate.
Let us now give some examples.
Example 8.3. Recall that a skew-symmetric integer matrix B is acyclic if the corre-
sponding directed graph (with an arrow i→ j associated with each entry bi,j > 0) has
no oriented cycles. If the matrix B(A) given by (7.1) is acyclic, then R〈〈A〉〉cyc = {0},
and so the only QP associated with A is (A, 0), which is clearly rigid.
Now suppose that A is 2-acyclic, and that B(A) is not necessarily acyclic but
is mutation equivalent to an acyclic matrix (i.e., can be transformed to an acyclic
matrix by a sequence of mutations). As a consequence of Corollary 6.11 and Theo-
rem 5.7, there exists a potential S such that (A, S) is a rigid reduced QP; moreover,
(A, S) is unique up to right-equivalences.
Example 8.4. For A arbitrary, the deformation space of a QP (A, 0) is naturally
identified with the space of potentials modulo cyclical equivalence, hence it is infinite-
dimensional provided A has at least one oriented cycle.
Example 8.5 (Cyclic triangle). Let Q be the quiver with three vertices 1, 2, 3 and
three arrows a : 1 → 2, b : 2 → 3 and c : 3 → 1:
An arbitrary potential S is cyclically equivalent to the one of the form S = F (cba),
where F ∈ K[[t]] is a formal power series. The deformation space Def(A, S) is
naturally isomorphic to the quotient space of tK[[t]] modulo the ideal generated by
QUIVERS WITH POTENTIALS I 33
tdF/dt. If charK = 0, and n ≥ 1 is the smallest exponent such that tn appears in F ,
then dimDef(A, S) = n− 1. In particular, (A, S) is rigid if and only if n = 1.
Now let us consider the QP (Ã, S̃) = µ̃2(A, S); in view of (5.6), (5.7) and (5.8), Ã
has four arrows a⋆, b⋆, c, [ba], and
S̃ = F (c[ba]) + [ba]a⋆b⋆.
Thus, if n ≥ 2 then (Ã, S̃) is reduced and so is equal to µ2(A, S) = (A, S). Since
µ2(A, S) has an oriented 2-cycle formed by the arrows c and [ba], the mutations at
vertices 1 and 3 cannot be applied. We see that the QP (A, F (cba)) is degenerate
for n ≥ 2.
Example 8.6 (Double cyclic triangle). Now consider the quiver with three vertices
1, 2, 3 and six arrows a1, a2 : 1 → 2, b1, b2 : 2 → 3 and c1, c2 : 3 → 1:
b2 ��=
Any potential S on A is cyclically equivalent to the one whose degree 3 component
belongs to the 8-dimensional space A31,1 = A1,3A3,2A2,1. It is known that the diagonal
action of the group GL32 on K
2 ⊗ K2 ⊗K2 has seven orbits, see e.g., [23, Chapter
14, Example 4.5]. Thus, by performing a change of arrows automorphism, we can
assume that the degree 3 component of S is one of the representatives of these orbits.
An easy case-by-case analysis shows that no potential can give rise to a rigid QP
on A.
For instance, let
(8.2) S = c1b1a1 + c2b2a2.
Then J(S) is the closure of the ideal in R〈〈A〉〉 generated by six elements
c1b1, b1a1, a1c1, c2b2, b2a2, a2c2.
One checks easily that the cyclic path c1b2a1c2b1a2 is not cyclically equivalent to an
element of J(S), hence (A, S) is not rigid.
Now let us compute µ2(A, S). Again setting (Ã, S̃) = µ̃2(A, S), we see that Ã has
ten arrows
a⋆1, a
2, c1, c2, [b1a1], [b1a2], [b2a1], [b2a2],
S̃ = c1[b1a1] + c2[b2a2] +
i,j=1
[biaj ]a
To obtain the splitting (4.5) of (Ã, S̃), we apply the automorphism ϕ of R〈〈Ã〉〉 fixing
all arrows except c1 and c2, and such that ϕ(ci) = ci−a
i . An easy check shows that
µ2(A, S) = (A, S) can be described as follows: A is 6-dimensional with the arrows
a⋆1, a
2, [b1a2], [b2a1], and
S = [b1a2]a
1 + [b2a1]a
34 HARM DERKSEN, JERZY WEYMAN, AND ANDREI ZELEVINSKY
Thus, the mutated QP (A, S) can be obtained from the initial QP (A, S) by a
renumbering of the vertices. This implies that one can apply to (A, S) unlimited
mutations at arbitrary vertices, so (A, S) is a non-rigid, nondegenerate QP.
Example 8.7. For each n ≥ 0, let us consider the following quiver Q(n), which we
refer to as the triangular grid of order n. The vertex set of Q(n) is
Q(n)0 = {(p, q, r) ∈ Z
≥0 | p+ q + r = n};
and there is a single arrow (p1, q1, r1) → (p2, q2, r2) if and only if (p2, q2, r2)−(p1, q1, r1)
is one of the three vectors (−1, 1, 0), (0,−1, 1), (1, 0,−1). Thus, the vertices of Q(n)
form a regular triangular grid with n2 cyclically oriented unit triangles. For example,
the quiver Q(4) is
<<yyyyyyyy
031oo
<<yyyyyyyy
121oo
<<yyyyyyyy
022oo
<<yyyyyyyy
211oo
<<yyyyyyyy
112oo
<<yyyyyyyy
013oo
<<yyyyyyyy
301oo
<<yyyyyyyy
202oo
<<yyyyyyyy
103oo
<<yyyyyyyy
004oo
Let A = A(n) be the arrow span of Q(n), and let a ∈ A (resp. b ∈ A, c ∈ A) denote
the sum of all arrows of Q(n) that are parallel translates of (−1, 1, 0) (resp. (0,−1, 1),
(1, 0,−1)). Thus, every interior vertex i has three incoming arrows eia, eib, eic and
three outgoing arrows aei, bei, cei. Every path of length d can be uniquely written
as ad · · · a2a1ej , where each as is one of the elements a, b, c, and j is a vertex; this
expression is non-zero if and only if the polygonal line obtained by attaching to the
vertex j the vectors corresponding to a1, a2, . . . , ad (in this order) is contained in our
grid.
Define a potential S ∈ A3 by setting
S = cba− bca.
Then the Jacobian ideal J(S) is generated by the elements
(cb− bc)ej , (ac− ca)ej , (ba− ab)ej
for all vertices j. It follows that the image of the path ad · · · a2a1ej in the Jacobian
algebra P(A, S) does not change under any permutation of the factors a1, . . . , ad.
In particular, we see that P(A, S) is spanned by the images of the paths ckbℓamej
for all vertices j and all k, ℓ,m such that 0 ≤ k, ℓ,m ≤ n; hence P(A, S) is finite-
dimensional. By a similar argument, it is easy to see that (A, S) is rigid. Indeed,
every potential on A is cyclically equivalent to an element of the closure of the span
of the elements (cba)mej for all vertices j and allm ≥ 0. Denoting by p the projection
QUIVERS WITH POTENTIALS I 35
R〈〈A〉〉 → Tr(P(A, S)), we see that the rigidity of (A, S) follows from the fact that
p(cbaej) = 0 for all j. Now if aej 6= 0 and h(aej) = i then we have
p(cbaej) = p(acbei) = p(cabei) = p(cbaei).
Continuing in the same way, we obtain that, for any m ≥ 1 such that amej 6= 0, we
have p(cbaej) = p(cbaek), where k is the end-point of the path a
mej. Taking m the
largest such that amej 6= 0, we conclude that p(cbaej) = 0, as desired.
As shown in [27], the quiver Q(3) in Example 8.7 is not mutation-equivalent to an
acyclic one. So there exist QPs with finite-dimensional Jacobian algebras (and also
rigid QPs), which are not mutation-equivalent to acyclic ones.
We now describe a procedure to obtain new QPs with finite-dimensional Jacobian
algebras (and new rigid QPs) from old ones.
Definition 8.8. For a QP (A, S) and a subset I of the vertex set Q0, we define the
restriction of (A, S) to I as the QP (A|I , S|I) given by
A|I =
i,j∈I
S|I = ψI(S),
where ψI : R〈〈A〉〉 → R〈〈A|I〉〉 is the algebra homomorphism such that ψI(a) = a
for a ∈ A|I , and ψI(b) = 0 for any arrow b not belonging to A|I .
Proposition 8.9. The homomorphism ψI induces an epimorphism of Jacobian alge-
bras P(A, S) → P(A|I , S|I) and an epimorphism of deformation spaces Def(A, S) →
Def(A|I , S|I). Therefore, if P(A, S) is finite-dimensional, or if (A, S) is rigid, then
the same is true for (A|I , S|I).
Proof. Remembering (3.1), it is easy to see that ψI(∂aS) = ∂aψI(S) for any arrow
a ∈ A|I , and ψI(∂bS) = 0 for any arrow b not belonging to A|I . Therefore, we have
ψI(J(S)) = J(ψI(S)), implying all the assertions. �
Corollary 8.10. Suppose that A and A′ are 2-acyclic, and there is a rigid QP (A, S)
on A. Let B(A) and B(A′) be the corresponding skew-symmetric integer matrices
given by (7.1). Suppose that B(A′) can be obtained by a simultaneous permutation
of rows and columns from some principal submatrix of a matrix mutation-equivalent
to B(A). Then there exists a rigid QP (A′, S ′) on A′.
Proof. In view of (7.1), the matrix B(A|I) is the principal submatrix of B(A) in-
volving rows and columns from I. Therefore, our assertion follows by combining
Proposition 8.9 with Corollary 6.11 and Proposition 7.1. �
We conclude this section with a combinatorial application of Corollary 8.10.
Corollary 8.11. Let B = B(A(n)) be the matrix associated with the triangular grid
of some order n (see Example 8.7). Then none of the matrices mutation-equivalent
to B contains
0 2 −2
−2 0 2
2 −2 0
36 HARM DERKSEN, JERZY WEYMAN, AND ANDREI ZELEVINSKY
as a principal submatrix.
Proof. Note that B′ = B(A′), where A′ is the quiver in Example 8.6. We have seen
that there exists a rigid QP on A, but not on A′. Thus our statement is a special
case of Corollary 8.10. �
Remark 8.12. Using the results in [1, Section 2.6] (cf. also [22, Theorem 1]), it
is easy to see that the quiver Q(n) in Example 8.7 is naturally associated with the
cluster algebra structure in the coordinate ring of the base affine space of the group
SLn+3. We have been informed by J. Schröer that, following his suggestion, A. Seven
has shown that a skew-symmetric matrix B′ associated with an arbitrary tree appears
as a principal submatrix in some matrix mutation-equivalent to the matrix B(A(n))
for some n. J. Schröer also informed us that Corollary 8.11 has been also proved by
B. Keller (using a different method).
9. Relation to cluster-tilted algebras
Let Q be a quiver with the vertex span R and the arrow span A. Assume that
Q is 2-acyclic. Let B(A) be the corresponding skew-symmetric integer matrix given
by (7.1). As shown in [19], the matrix B(A) gives rise to a cluster algebra of finite
type if and only if Q is mutation-equivalent to a Dynkin quiver (that is, an arbitrary
orientation of a Dynkin diagram of one of the types An, Dn, E6, E7, or E8). In
particular, as we have seen in Example 8.3, there is a rigid reduced QP (A, S), and
it is unique up to a right-equivalence; in fact, (the right-equivalence class of) (A, S)
is obtained by a sequence of mutations from a QP (A0, 0), where A0 is associated to
a Dynkin quiver. We now present an explicit choice of such a potential S. To do
this, we need some preparation.
First note that, according to [19, Theorem 1.8], every quiver mutation-equivalent
to a Dynkin quiver has no multiple edges, that is |bi,j| ≤ 1 for every entry of B(A).
In other words, we have:
(9.1) dimAi,j ≤ 1 for all i and j.
Therefore, we can unambiguously denote by ai,j the only arrow in a non-zero subspace
Ai,j . We will also use the convention that ai,j = 0 whenever Ai,j = {0}.
Second, we use the following terminology: a chordless cycle in (the underlying
graph of) Q is a d-subset of vertices I ⊆ Q0 that can be bijectively labeled by the
elements of Z/dZ so that the edges between them are precisely {i, i+1} for i ∈ Z/dZ.
According to [19, Proposition 9.7], if Q is mutation-equivalent to a Dynkin quiver
then the arrows of every chordless cycle in Q must be cyclically oriented. In terms
of the arrow span A, this condition can be stated as follows:
For any chordless d-cycle I, there exists a bijection ν : Z/dZ → I such(9.2)
that the set of arrows in the restriction A|I is {aν(i),ν(i+1) | i ∈ Z/dZ}.
(see Definition 8.8). Note that the choice of a bijection ν in (9.2) is unique up to a
cyclic shift.
We call a potential S on A primitive if it has the form
(9.3) S =
xIaν(1),ν(2) · · · aν(d−1),ν(d)aν(d),ν(1),
QUIVERS WITH POTENTIALS I 37
where the (finite) sum is over all chordless cycles I in Q, for each I there is a
bijection ν chosen as in (9.2), and the coefficients xI are some non-zero elements of
the base field K.
Proposition 9.1. If a quiver Q with the arrow span A is mutation-equivalent to a
Dynkin quiver, and S is a primitive potential on A, then the QP (A, S) is rigid.
To prove Proposition 9.1, we impose on Q a weaker assumption that its arrow
span A satisfies (4.13), (9.1) and (9.2). Choose some vertex k, and (as in Section 7)
let µk(A) denote the 2-acyclic R-bimodule such that the skew-symmetric matrix
B(µk(A)) is obtained from B(A) by the mutation at k. It is easy to see that µk(A)
satisfies (9.1) but not necessarily (9.2). In view of Corollary 6.11, the assertion of
Proposition 9.1 is a consequence of the following lemma.
Lemma 9.2. Suppose that the arrow span A of a quiver Q satisfies (4.13), (9.1) and
(9.2), and that µk(A) also satisfies (9.2) for some vertex k. Let S be a primitive
potential on A. Then the QP (A, S) = µk(A, S) is right-equivalent to a QP on µk(A)
with a primitive potential.
Proof. Let µ̃k(A, S) = (Ã, S̃) be the QP given by (5.6), (5.7) and (5.8). Denote by
In(k) (resp. Out(k)) the set of vertices j (resp. i) such that dimAk,j = 1 (resp.
dimAi,k = 1). In view of (9.2), every arrow a with both ends in In(k) ∪Out(k) has
h(a) ∈ In(k) and t(a) ∈ Out(k). We denote the set of these arrows by Q′1. The
arrows of Ã can be unambiguously denoted as follows:
• ãi,j = ai,j for all i, j different from k and such that ai,j 6= 0.
• ãi,j = [ai,kak,j] for all i ∈ Out(k), j ∈ In(k).
• ãj,k = a
k,j for j ∈ In(k).
• ãk,i = a
i,k for i ∈ Out(k).
We can split S into the sum of four terms
(9.4) S = S1 + S2 + S3 + S4,
where
• S1 involves (oriented) 3-cycles ai,kak,jaj,i.
• S2 involves chordless d-cycles through k with d ≥ 4;
• S3 involves chordless cycles having an arrow aj,i ∈ Q
1 but not passing
through k;
• S4 involves chordless cycles having no arrows with both ends in In(k)∪{k}∪
Out(k).
Using (9.2), it is easy to see that every chordless cycle I involved in S2 or S3 has
exactly one common point with each of the sets In(k) and Out(k). Also note that
every term in S1 or S3 contains exactly one arrow from Q
1, while none of the terms
in S2 or S4 contain such arrows. Remembering (5.8), we write the potential S̃ as
follows:
(9.5) S̃ = [S1] + [S2] + [S3] + [S4] + ∆k.
We have
[S1] =
aj,i∈Q
x{i,j,k}ãi,jaj,i,
38 HARM DERKSEN, JERZY WEYMAN, AND ANDREI ZELEVINSKY
and this is the degree 2 component S̃(2) of S̃. In view of (5.16), the arrows in A are
obtained from those in Ã by removing all arrows aj,i ∈ Q
1 and their opposites ãi,j.
Inspecting the other terms in (9.5), it is easy to see that
[S2] = S3, [S4] = S4,
∆k = S1 +
aj,i∈Q
ãi,jãj,kãk,i,
[S3] =
aj,i∈Q
fi,jaj,i,
where fi,j = ∂aj,iS3, and the terms S1, S3 and S4 have the same meaning as in (9.4)
with A replaced by A. Let ϕ be the automorphism of R〈〈Ã〉〉 acting on arrows as
follows:
ϕ(aj,i) = x
{i,j,k}
(aj,i − ãj,kãk,i), ϕ(ãi,j) = ãi,j − x
{i,j,k}
for every aj,i ∈ Q
1, and ϕ fixes the rest of the arrows. Then we have
ϕ(S̃) = S1 + S3 + S4 + ϕ(
aj,i∈Q
(x{i,j,k}ãi,jaj,i + ãi,j ãj,kãk,i + fi,jaj,i))
= S1 + S3 + S4 +
aj,i∈Q
(ãi,jaj,i − x
{i,j,k}
fi,jãj,kãk,i).
We see that the degree 2 component of ϕ(S̃) is
ϕ(S̃)(2) =
aj,i∈Q
ãi,jaj,i,
and a moment’s reflection shows that
aj,i∈Q
{i,j,k}
fi,jãj,kãk,i
can be viewed as the component S2 of a primitive potential on A. We conclude that
µk(A, S) is right-equivalent to (A, S1 + S2 + S3 + S4), finishing the proof. �
We conclude this section by briefly discussing a connection between Jacobian al-
gebras of rigid QPs and cluster-tilted algebras introduced in [8]. We refer to [8] for
precise definitions; roughly speaking, cluster-tilted algebras are endomorphism rings
of tilting objects in cluster categories. One can associate such an algebra to any
quiver Q which is mutation-equivalent to a Dynkin quiver. As shown in [13, Theo-
rem 4.1] (for type A) and [9, Theorems 4.1, 4.2], the cluster-tilted algebra associated
to Q is isomorphic to the path algebra of Q modulo an explicitly described ideal of
relations. By inspection, this ideal is exactly the Jacobian ideal of a primitive poten-
tial S given by (9.3). Thus, Proposition 9.1 has the following corollary, which shows
that the Jacobian algebras of QPs can be viewed as generalizations of cluster-tilted
algebras.
Corollary 9.3. If a quiver Q with the arrow span A is mutation-equivalent to a
Dynkin quiver, then the Jacobian algebra P(A, S) of a rigid QP on A (explicitly
given by (9.3)) is isomorphic to the cluster-tilted algebra associated to Q.
QUIVERS WITH POTENTIALS I 39
10. Decorated representations and their mutations
The following definition is inspired by [28].
Definition 10.1. A decorated representation of a QP (A, S) is a pair M = (M,V ),
where V is a finite-dimensional (left) R-module, and M is a finite-dimensional
R〈〈A〉〉-module annihilated by J(S).
Equivalently, M is a finite-dimensional P(A, S)-module. We will sometimes write
M = (A, S,M, V ) and refer to such a quadruple as a QP-representation.
We have M =
Mi and V =
Vi, where Mi = eiM and Vi = eiV . With
some abuse of notation, for u ∈ R〈〈A〉〉 or u ∈ P(A, S), we denote the multiplication
operator m 7→ um on M simply as u : M → M ; we write u = uM if we need
to emphasize the dependence of this operator on M . In particular, for each arrow
a ∈ A, we have a :Mt(a) → Mh(a), and a|Mi = 0 for i 6= t(a).
Note that every finite-dimensional R〈〈A〉〉-module M is nilpotent, i.e., M is anni-
hilated by mn for n≫ 0. We thank Bill Crawley-Boevey for pointing this out to us;
the following argument was also suggested by him. The above claim is a special case
of the following more general fact: if a ring U with unit is complete in the I-adic
topology for some two-sided ideal I, and M is a U -module of finite length n, then
InM = {0}. Indeed, the element 1 + x is invertible in U for any x ∈ I, since it has
inverse 1 − x + x2 − x3 + · · · . Thus I is contained in the Jacobson radical J (since
J is the set of x ∈ U such that 1 + xy is invertible for all y ∈ U). Thus IS = {0}
for any simple U -module S (since J is the intersection of annihilators of all simple
modules). Now if M has composition series
{0} =M0 ⊂M1 ⊂ · · · ⊂Mn =M,
then for all i ≥ 1, we have I(Mi/Mi−1) = {0}, so IMi ⊆ Mi−1. It follows that
InM = {0}, as claimed.
The aim of this section is to extend the definition of QP-mutations in Corollary 5.4
and Definition 5.5 to the level of QP-representations, and to prove a representation-
theoretic extension of Theorem 5.7. To do this, we first introduce right-equivalence
for QP-representations.
Definition 10.2. Let (A, S) and (A′, S ′) be QPs on the same set of vertices, and let
M = (M,V ) (resp. M′ = (M ′, V ′)) be a decorated representation of (A, S) (resp.
of (A′, S ′)). A right-equivalence between M and M′ is a triple (ϕ, ψ, η), where:
• ϕ : R〈〈A〉〉 → R〈〈A′〉〉 is a right-equivalence between (A, S) and (A′, S ′) (see
Definition 4.2);
• ψ : M → M ′ is a vector space isomorphism such that ψ ◦ uM = ϕ(u)M ′ ◦ ψ
for u ∈ R〈〈A〉〉;
• η : V → V ′ is an isomorphism of R-modules.
Remark 10.3. The usual notion of isomorphism of decorated representations M =
(M,V ) and M′ = (M ′, V ′) of the same QP (A, S) (namely, that M and M ′ are iso-
morphic P(A, S)-modules, and V and V ′ are isomorphic R-modules) is a special case
of right-equivalence with the ϕ-component being the identity. The right-equivalence
seems to be more relevant for applications to cluster algebras. To illustrate, consider
the Kronecker quiver Q with two vertices 1 and 2, and two arrows a and b from
40 HARM DERKSEN, JERZY WEYMAN, AND ANDREI ZELEVINSKY
1 to 2. Since Q has no oriented cycles, it has only one QP (A, S): the one with
S = 0. Thus, a decorated representation M = (M,V ) with V = 0 is just a usual
representation of the quiver Q, i.e., it consists of two vector spaces M1 and M2, and
two linear maps a and b from M1 to M2. Such representations were classified by
Kronecker. In particular, he proved that, for every n ≥ 1, the isomorphism classes of
indecomposable Q-representations with dimM1 = dimM2 = n are naturally param-
eterized by the projective line. However all these representations are right-equivalent
to each other. This is more in sync with the fact that, as discussed in [16], all these
representations give rise to the same element of the corresponding cluster algebra.
We now present a representation-theoretic extension of Theorem 4.6. Let M =
(A, S,M, V ) be a QP-representation, and let ϕ : R〈〈Ared⊕C〉〉 → R〈〈A〉〉 be a right-
equivalence of the QPs (Ared, Sred)⊕(C, T ) and (A, S), where (Ared, Sred) is a reduced
QP, and (C, T ) is a trivial QP, see Theorem 4.6. We define a R〈〈Ared〉〉-module M
by setting M ′ =M as a K-vector space with the action of R〈〈Ared〉〉 given by uM ′ =
ϕ(u)M . In view of Proposition 4.5, this makes a quadruple Mred = (Ared, Sred,M
′, V )
a well-defined QP-representation.
Definition 10.4. We call the QP-representation Mred given by the above construc-
tion the reduced part of M.
This terminology is justified by the following.
Proposition 10.5. The right-equivalence class of Mred is determined by the right-
equivalence class of M.
Proof. To get more in sync with the notation in Proposition 4.9, we change our
notation a little bit and assume that M is a decorated representation of a QP
(A ⊕ C, S + T ), where (A, S) is a reduced QP, and (C, T ) a trivial one. Let ϕ
be an auto-right-equivalence of (A ⊕ C, S + T ), that is, an algebra automorphism
of R〈〈A ⊕ C〉〉 such that ϕ(S + T ) is cyclically equivalent to S + T . To prove
Proposition 10.5, it suffices to show the following:
there exists an algebra automorphism ϕ′ of R〈〈A〉〉 such that(10.1)
ϕ′(S) is cyclically equivalent to S, and ϕ′(u)M = ϕ(u)M for u ∈ R〈〈A〉〉.
As in the proof of Proposition 4.9, let L denote the closure of the two-sided ideal
in R〈〈A⊕ C〉〉 generated by C. Recall from (4.10) that we have
R〈〈A⊕ C〉〉 = R〈〈A〉〉 ⊕ L,
J(S + T ) = J(S)⊕ L
(in the last equality, J(S) is understood as the Jacobian ideal of S in R〈〈A〉〉). In
particular, we have uM = 0 for u ∈ L.
Let ψ : R〈〈A〉〉 → R〈〈A〉〉 denote the restriction to R〈〈A〉〉 of the composition
p ◦ ϕ, where p is the projection of R〈〈A⊕ C〉〉 onto R〈〈A〉〉 along L. Then we have
ψ(u)M = ϕ(u)M for u ∈ R〈〈A〉〉.
According to (4.12), ψ is an algebra automorphism of R〈〈A〉〉. Furthermore, using
(4.12) in conjunction with Proposition 4.10, we conclude that J(ψ(S)) = J(S), and
QUIVERS WITH POTENTIALS I 41
that there exists an algebra automorphism η of R〈〈A〉〉 such that η(ψ(S)) is cyclically
equivalent to S, and η(u)− u ∈ J(S) for u ∈ R〈〈A〉〉. Taking ϕ′ = η ◦ψ, we see that
ϕ′(u)M = η(ψ(u))M = ψ(u)M = ϕ(u)M
for u ∈ R〈〈A〉〉. Thus ϕ′ satisfies all the conditions in (10.1), and we are done. �
We turn to the definition of mutations for a QP-representation M = (A, S,M, V ).
We fix a vertex k satisfying (5.1), and let a1, . . . , as be all arrows with h(ap) = k,
and b1, . . . , bt be all arrows with t(bq) = k. We denote
(10.2) Min =
Mt(ap), Mout =
Mh(bq).
Let α = αM : Min → Mk and β = βM : Mk → Mout be K-linear maps given in the
matrix form by
(10.3) α =
a1 a2 · · · as
, β =
 .
We also introduce a K-linear map γ = γM : Mout → Min as follows. Replacing S
if necessary by a cyclically equivalent potential, we may assume that S ∈ R〈〈A〉〉k̂,k̂
(see (6.1)). We identify R〈〈A〉〉
k̂,k̂
with R〈〈Ã
k̂,k̂
〉〉 as in Lemma 6.2. This allows us
to define the component γp,q :Mh(bq) →Mt(ap) of γ by setting
(10.4) γp,q = ∂[bqap]S.
Thus, we have constructed the following triangle of linear maps:
(10.5) Mk
<<zzzzzzzz
Lemma 10.6. We have αγ = 0 and γβ = 0.
Proof. The q-th component of αγ is equal to
ap∂[bqap]S = ∂bqS;
therefore, αγ = 0, since M is a representation of P(A, S). Similarly, the p-th com-
ponent of γβ is equal to ∑
(∂[bqap]S)bq = ∂apS,
implying that γβ = 0. �
Now let (Ã, S̃) be given by (5.4) and (5.8). We associate to a QP-representation
M = (A, S,M, V ) the QP-representation µ̃k(M) = (Ã, S̃,M, V ) defined as follows.
First, we set
(10.6) M i =Mi, V i = Vi (i 6= k).
42 HARM DERKSEN, JERZY WEYMAN, AND ANDREI ZELEVINSKY
We define Mk and V k by
(10.7) Mk =
ker γ
⊕ im γ ⊕
⊕ Vk, V k =
ker β
ker β ∩ im α
We now define the action on M of all arrows in Ã (recall that they are given
by (5.6) and (5.7)). Thus, for every such arrow c, we need to define a linear map
cM :M t(c) → Mh(c).
First, we set
cM = cM
for every arrow c not incident to k, and
[bqap]M = (bqap)M
for all p and q.
To define the action of the remaining arrows a⋆p and b
q , we assemble them into the
operators
b⋆1 b
2 · · · b
in the same way as in (10.3). Thus, we need to define linear maps
α :Mout =M in →Mk, β :Mk → Mout =Min.
We will use the following notational convention: whenever we have a pair U1 ⊆ U2
of vector spaces, denote by ι : U1 → U2 the inclusion map, and by π : U2 → U2/U1
the natural projection. We now introduce the following splitting data:
Choose a linear map ρ :Mout → ker γ such that ρι = idker γ.(10.8)
Choose a linear map σ : kerα/im γ → kerα such that πσ = idkerα/im γ .(10.9)
Then we define:
(10.10) α =
 , β =
0 ι ισ 0
Having defined the action of all arrows in Ã on M , we can view M as a module
over the path algebra R〈Ã〉. The property thatM is annihilated by m(A)n for n≫ 0
implies thatM is annihilated by Ãn for n≫ 0. This allows us to viewM as a module
over the completed path algebra R〈〈Ã〉〉.
Proposition 10.7. The above definitions make µ̃k(M) = (M,V ) a decorated repre-
sentation of (Ã, S̃).
QUIVERS WITH POTENTIALS I 43
Proof. We only need to show that (∂cS̃)M = 0 for every arrow c in Ã. If c is not
incident to k, the desired statement follows from (6.6). If c is one of the arrows [bqap],
then, in view of (6.5) and (10.4), it is enough to show that
q + γp,q = 0
for all p and q. In other words, we need to show that βα = −γ; But this follows at
once by multiplying the row and column given by (10.10).
It remains to show that (∂a⋆p S̃)M = 0 and (∂b⋆q S̃)M = 0 for all p and q. We first
deal with the first equality. Remembering (5.8) and (5.9), we see that
(∂a⋆p S̃)M = (
b⋆q[bqap])M = (
(b⋆q)M(bq)M)(ap)M .
Thus it suffices to show that
(b⋆q)M(bq)M = 0,
or equivalently, αβ = 0. In view of (10.10), we have
 = 0,
as desired (the equality πρβ = 0 is immediate from the definitions, while γβ = 0 by
Lemma 10.6).
The remaining equality (∂b⋆ν S̃)M = 0 is proved in a similar way. We have
(∂b⋆q S̃)M = (
[bqap]a
p)M = (bq)M
(ap)M(a
p)M .
Thus, it suffices to observe that
0 αι αισ 0
Example 10.8. Let us illustrate the definition of the operation µ̃k by a special case
where the vertex k is a sink or a source. First suppose k is a sink, that is, there are
no arrows b with t(b) = k. Then we have Mout = {0}, hence β = 0 and γ = 0. Thus,
(10.7) simplifies to
Mk = kerα⊕ Vk, V k = coker α .
The arrow span Ã is obtained from A by reversing all arrows a with h(a) = k, that
is, replacing every such arrow a with a⋆. Thus, k becomes a source for Ã, hence
M in = {0} and α = 0. The choice of splitting data (10.8) and (10.9) becomes
immaterial, and the second equality in (10.10) simplifies to
Note that we have Ã
k̂,k̂
k̂,k̂
, and the potential S̃ ∈ R〈〈Ã
k̂,k̂
〉〉 is naturally identified
with S.
44 HARM DERKSEN, JERZY WEYMAN, AND ANDREI ZELEVINSKY
The case where k is a source is completely similar. In this case we haveMin = {0},
hence α = 0 and γ = 0. It follows that
Mk = coker β ⊕ Vk, V k = ker β ,
and the map α :M in =Mout →Mk is given by
In both cases, µ̃k coincides with the “extended reflection functor” introduced in
[28]; furthermore, if we ignore the decorations (and the potentials), it becomes the
classical Bernstein-Gelfand-Ponomarev reflection functor at k, see [3].
Now we return to the case of an arbitrary vertex k.
Proposition 10.9. The isomorphism class of the decorated representation µ̃k(M)
does not depend on the choice of the splitting data (10.8) – (10.9).
Proof. We have the following freedom in choosing the splitting data: one can replace
ρ :Mout → ker γ with ρ
′ = ρ+ ξγ for some linear map ξ : im γ → ker γ, and replace
σ : kerα/im γ → kerα by σ′ = σ + η for some linear map η : kerα/im γ → im γ.
Let α′ and β
be the maps obtained by replacing ρ with ρ′, and σ with σ′ in (10.10).
It is enough to show that ψα = α′ and β
ψ = β for some linear automorphism
ψ : Mk → Mk. Decomposing Mk as in (10.7), we define ψ as the block-triangular
matrix
I πξ 0 0
0 I −η 0
0 0 I 0
0 0 0 I
 ,
where I stands for the identity transformation. The invertibility of ψ is obvious,
and the desired equalities ψα = α′ and β
ψ = β are checked by direct matrix
multiplication. �
Proposition 10.10. The right-equivalence class of the representation µ̃k(M) is de-
termined by the right-equivalence class of M.
Proof. Let ϕ be an automorphism of R〈〈A〉〉, and let M′ = (A,ϕ(S),M ′, V ′) be the
QP-representation defined as follows: V ′ = V and M ′ =M as R-modules, while the
R〈〈A〉〉-actions in M and M ′ are related by
(10.11) uM = ϕ(u)M ′ (u ∈ R〈〈A〉〉)
(note that (10.11) indeed defines a representation of (A,ϕ(S)) in view of Proposi-
tion 3.7). To prove Proposition 10.10, it suffices to show that the representations
µ̃k(M) and µ̃k(M
′) are right-equivalent.
We retain all the above notation related to M and µ̃k(M); in particular, α, β
and γ stand for the linear maps in the triangle (10.5). Let α′, β ′ and γ′ denote the
corresponding maps for the representation M′. Our first order of business is to relate
these maps to α, β and γ.
We can write the action of ϕ on the arrows a1, . . . , as as follows:
(10.12)
ϕ(a1) ϕ(a2) · · · ϕ(as)
a1 a2 · · · as
QUIVERS WITH POTENTIALS I 45
where C = C0 +C1 is an invertible s× s matrix as in (5.14). Similarly, the action of
ϕ on the arrows b1, . . . , bs can be written as
(10.13)
ϕ(b1)
ϕ(b2)
ϕ(bt)
 = D
 ,
where D = D0 +D1 is an invertible t× t matrix as in (5.15). Therefore we have
a1 a2 · · · as
ϕ(a1) ϕ(a2) · · · ϕ(as)
(10.14)
a1 a2 · · · as
C)M ′ = α
′CM ′ ,
and similarly,
(10.15) β = DM ′β
here CM ′ (resp. DM ′) is understood as an R-module automorphism of M
in = Min
(resp. of M ′out =Mout).
Turning to the maps γ and γ′, we claim that they are related by
(10.16) γ′ = CM ′γDM ′.
To see this, we use (10.4) and (3.6) to write
γ′p,q = (∂[bqap]ϕ(S))M ′(10.17)
∆[bqap](ϕ(c))�ϕ(∂cS))M ′,
where the sum is over all arrows c in Ãk̂,k̂. If c is one of the arrows in A then by
(10.11) we have
ϕ(∂cS)M ′ = (∂cS)M = 0;
remembering the definition (3.3), we see that c does not contribute to (10.17). Thus,
we have
(10.18) γ′p,q = (
p′,q′
∆[bqap](ϕ(bq′ap′))�ϕ(∂[bq′ap′ ]S))M ′.
Remembering (3.2), and using (10.12) and (10.13), we see that the summand with
(p′, q′) = (p, q) in (10.18) contains among its terms the (p, q)-entry of the matrix
(Cϕ(∂[bqap]S)D)M ′ = CM ′γDM ′.
Thus, to prove (10.16), it remains to show that the rest of the terms in (10.18) add
up to 0. Again using the definitions (3.2) and (3.3), we can rewrite the rest of the
sum in (10.18) as S1 + S2, where
S1 = (
p′,q′
∆[bqap](ϕ(bq′))�ϕ(ap′ · ∂[bq′ap′ ]S))M ′,
S2 = (
p′,q′
∆[bqap](ϕ(ap′))�ϕ(∂[bq′ap′ ]S · bq′))M ′.
46 HARM DERKSEN, JERZY WEYMAN, AND ANDREI ZELEVINSKY
It remains to observe that
S1 = (
∆[bqap](ϕ(bq′))�ϕ(∂bq′S))M ′ = 0
since ϕ(∂bq′S)M ′ = (∂bq′S)M = 0; and similarly,
S2 = (
∆[bqap](ϕ(ap′))�ϕ(∂ap′S))M ′ = 0.
In view of (10.14), (10.15) and (10.16), we have
kerα = C−1M ′(kerα
′), im α = im α′,
ker β = ker β ′, im β = DM ′(im β
′),(10.19)
ker γ = DM ′(ker γ
′), im γ = C−1M ′(im γ
Recall that the spaces M and V in the decorated representation µ̃k(M) = (M,V )
of (Ã, S̃) are given by (10.6) and (10.7). We express the decorated representation
µ̃k(M
′) = (M ′, V ′) of (Ã, ϕ̃(S)) in the same way, with the maps α, β and γ replaced
by α′, β ′ and γ′. In particular, we have V ′ = V , and M ′i = M i = Mi for i 6= k.
To specify the actions of R〈〈A〉〉 in M and M ′, we need to choose the splitting data
(ρ, σ) and (ρ′, σ′) as in (10.8) and (10.9). Note that, in view of (10.19), we can choose
(10.20) ρ′ = D−1M ′ρDM ′, σ
′ = CM ′σC
here with some abuse of notation we use the same notation C−1M ′ for the isomorphism
kerα′ → kerα and the induced isomorphism kerα′/im γ′ → kerα/im γ.
Everything is now in place for defining the desired right-equivalence (ϕ̂, ψ, η) be-
tween µ̃k(M) and µ̃k(M
′) (see Definition 10.2). First of all, we define ϕ̂ : R〈〈Ã〉〉 →
R〈〈Ã〉〉 as the right-equivalence between (Ã, S̃) and (Ã, ϕ̃(S) constructed in the proof
of Lemma 5.3. In particular, we have
ϕ̂(a⋆1)
ϕ̂(a⋆2)
ϕ̂(a⋆s)
 = C
 ,
ϕ̂(b⋆1) ϕ̂(b
2) · · · ϕ̂(b
b⋆1 b
2 · · · b
Next we define ψ :M →M ′ as the identity map on ⊕i 6=kM i = ⊕i 6=kMi = ⊕i 6=kM ′i,
and the restriction ψ|Mk :Mk →M
k given by the block-diagonal matrix
(10.21) ψ|Mk =
D−1M ′ 0 0 0
0 CM ′ 0 0
0 0 CM ′ 0
0 0 0 I
(this is well-defined in view of (10.19)). Finally, we define η : V → V ′ simply as the
identity map.
The only thing to check is the equality ψ ◦ cM = ϕ̂(c)M ′ ◦ ψ for any arrow c ∈ Ã.
And the only case that may require some consideration is when c is one of the arrows
a⋆p or b
q . Unraveling the definitions, it suffices to show that
β = C−1M ′β
′ ◦ ψ|Mk , ψ|Mk ◦ α = α
′D−1M ′.
QUIVERS WITH POTENTIALS I 47
But this is an immediate consequence of the definitions (10.21) and (10.10) (we also
need an analogue of (10.10) for the maps β ′ and α′, using the splitting data (10.20)).
This completes the proof of Proposition 10.10. �
Note that in the above treatment of the operation M 7→ µ̃k(M) for a QP-
representation M = (A, S,M, V ), the QP (A, S) was not assumed to be reduced.
Recall from Proposition 10.5 that we have a well-defined operation M 7→ Mred on
(right-equivalence classes of) QP-representations. The following property is imme-
diate from definitions.
Proposition 10.11. Let (A, S) be a QP satisfying (5.1). Then, for every represen-
tation M of (A, S), the representation µ̃k(M)red is right-equivalent to µ̃k(Mred)red.
Recall that, according to Corollary 5.4 and Definition 5.5, the correspondence µ̃k :
(A, S) 7→ µ̃k(A, S) = (Ã, S̃) gives rise to the mutation (A, S) 7→ µk(A, S) = (A, S),
which is a well-defined bijective transformation on the set of right-equivalence classes
of reduced QPs satisfying (5.1). Here (A, S) is the reduced part of (Ã, S̃). Now for
every QP-representation M = (A, S,M, V ) of a reduced QP (A, S) we define
(10.22) µk(M) = µ̃k(M)red;
thus, µk(M) is a decorated representation (A, S,M, V ) of a reduced QP (A, S).
Combining Propositions 10.5 and 10.10, we obtain the following important corollary.
Corollary 10.12. The correspondence M 7→ µk(M) is a well-defined transforma-
tion on the set of right-equivalence classes of decorated representations of reduced
QPs satisfying (5.1).
We refer to the transformation M 7→ µk(M) in Corollary 10.12 as the mutation at
vertex k. With some abuse of terminology, we will talk about mutations of decorated
representations (rather than their right-equivalence classes).
The following result naturally extends Theorem 5.7.
Theorem 10.13. The mutation µk of decorated representations is an involution; that
is, for every decorated representation M of a reduced QP (A, S) satisfying (5.1), the
decorated representation µ2k(M) of a QP µ
k(A, S) is right-equivalent to M.
Proof. In view of Proposition 10.11, µ2k(M) is right-equivalent to µ̃
k(M)red. There-
fore, is suffices to show that µ̃2k(M)red is right-equivalent to M.
We write the QP-representation µ̃2k(M) as µ̃
k(M) = (
S,M, V ). The QP (
is given by (5.18) and (5.19). In particular, there is a natural embedding of A into
A identifying A with the reduced part
Ared. Furthermore, as shown in the proof of
Theorem 5.7, an automorphism of R〈〈
A〉〉 that establishes the right-equivalence in
(5.17) can be chosen so that it restricts to an automorphism ϕ0 : R〈〈A〉〉 → R〈〈A〉〉
acting as follows:
ϕ0 multiplies each of the arrows b1, . . . , bt by −1,(10.23)
and fixes the rest of the arrows in A.
48 HARM DERKSEN, JERZY WEYMAN, AND ANDREI ZELEVINSKY
In view of Definition 10.4, the QP-representation µ2k(M) = µ̃
k(M)red can be realized
as (A, S,M ′, V ′), where M ′ = M and V ′ = V as vector spaces, and the action of
R〈〈A〉〉 in M ′ is given by
(10.24) uM ′ = ϕ0(u)M (u ∈ R〈〈A〉〉).
To prove Theorem 10.13, it suffices to show that the decorated representation (M ′, V ′)
of (A, S) is isomorphic to (M,V ).
We first compute M ′ = M and V ′ = V as vector spaces. According to (10.6), we
M ′i =M i =M i =Mi, V
i = V i = V i = Vi
for all i 6= k. As for the spaces M ′k and V
k , they are given as in (10.7), with the maps
α, β, and γ replaced by α, β, and γ, respectively. Recall that α and β are given by
(10.10). As for γ, by applying the definition (10.4) to the potential S̃ given by (5.8)
and (5.9), we see that
(10.25) γ = βα.
As a direct consequence of the definitions, we conclude that
kerα = im β, im α = ker γ
⊕ im γ ⊕ {0} ⊕ {0},
ker β = ker γ
⊕ {0} ⊕ {0} ⊕ Vk, im β = kerα,(10.26)
ker γ = ker(βα), im γ = im (βα).
It follows that
V ′k = V k =
ker β
ker β ∩ im α
= Vk,
and so V ′ = V , as desired. We also have
M ′k =
ker(βα)
⊕ im (βα)⊕
im (βα)
ker β
ker β ∩ im α
We now make the following easy observations:
• the map α induces an isomorphism ker(βα)/ kerα → ker β ∩ im α;
• the map β induces an isomorphism im α/(ker β ∩ im α) → im (βα);
• the map β induces an isomorphism Mk/(ker β + im α) → im β/im (βα).
• there is a natural isomorphism ker β/(ker β ∩ im α) → (ker β + im α)/im α.
Using these isomorphisms, we can identify M ′k with the space
(10.27) M ′k = (ker β ∩ im α)⊕
ker β ∩ im α
ker β + im α
ker β + im α
To describe the action of R〈〈A〉〉 in M ′, we need only to describe the maps α′ :
Min → M
k and β
′ : M ′k → Mout constructed in the same way as in (10.3). As in
(10.10), the definition of α′ and β ′ involves splitting data (10.8) - (10.9). In view of
(10.26), in the current situation the splitting data take the following form:
Choose a linear map ρ :Min → ker(βα) such that ρι = idker(βα).
Choose a linear map σ : im β/im (βα) → im β such that πσ = idim β/im (βα).
QUIVERS WITH POTENTIALS I 49
Adapting (10.10) to the current situation (in particular, realizing M ′k as in (10.27)),
we see that the maps α′ and β ′ take the following form:
(10.28) α′ =
 , β
0 −β −ισβ 0
here with some abuse of notation we denote by the same symbol β the two maps
im α/ kerβ ∩ im α → Mout and Mk/(ker β + im α) → im β/im (βα) induced by β.
Note that the appearance of the minus sign in β ′ is caused by the minus sign in
(10.23).
To complete the proof of Theorem 10.13, it remains to construct an isomorphism
of vector spaces ψ :M ′k →Mk such that
(10.29) ψα′ = α, βψ = β ′.
To do this, notice that the four direct summands in (10.27) are the factors in the
filtration
{0} ⊆ ker β ∩ im α ⊆ im α ⊆ ker β + im α ⊆Mk.
Choose some sections
σ1 : im α/(ker β ∩ im α) → im α,
σ2 : (ker β + im α)/im α→ (ker β + im α),
σ3 :Mk/(ker β + im α) →Mk
for the three factors of this filtration, so that they satisfy:
im σ1 = α(ker ρ), im σ2 ⊆ ker β, im (βσ3) = im σ.
Now define an an isomorphism ψ :M ′k →Mk by setting
−ι −ισ1 −ισ3 −ισ2
The equalities (10.29) are checked by a direct inspection, finishing the proof. �
Note that there is an obvious way to define direct sums for decorated repre-
sentations of a given QP (A, S). Hence we can talk about indecomposable QP-
representations. Clearly, the right-equivalence relation respects direct sums and in-
decomposability. It is also immediate from the definitions that any mutation µk of
QP-representations sends direct sums to direct sums. Combining this with Theo-
rem 10.13, we obtain the following corollary.
Corollary 10.14. Any mutation µk is an involution on the set of right-equivalence
classes of indecomposable decorated representations of reduced QPs satisfying (5.1).
We call a QP-representation (A, S,M, V ) positive if V = {0}. Thus, indecompos-
able positive representations are just indecomposable P(A, S)-modules. In partic-
ular, for every vertex k, the simple representation Sk(A, S) is the indecomposable
positive representation of (A, S) such that dimMi = δi,k. We denote by S
k (A, S) the
indecomposable representation (M,V ) of (A, S) such thatM = {0} and dim Vi = δi,k.
We refer to S−k (A, S) as the negative simple representation at k. The following propo-
sition is immediate from the definitions.
50 HARM DERKSEN, JERZY WEYMAN, AND ANDREI ZELEVINSKY
Proposition 10.15. Any indecomposable QP-representation is either positive, or
negative simple. If µk(A, S) = (A, S) then we have
(10.30) µk(Sk(A, S)) = S
k (A, S), µk(S
k (A, S)) = Sk(A, S);
and this is the only mutation that interchanges positive and negative indecomposable
representations.
11. Some three-vertex examples
In this section we illustrate the action of mutations on QP-representations by
some examples dealing with three-vertex quivers. All the representations (M,V )
considered below will be positive, i.e., V = {0}.
Example 11.1. Let Q be the quiver with three vertices 1, 2, 3 and two arrows a :
1 → 2 and b : 2 → 3:
Since Q is acyclic, the only QP on it is (A, 0). We have µ2(A, 0) = (A, S), where A
is the arrow span of the quiver Q given by
and S = b⋆[ba]a⋆. Thus, positive representations of (A, 0) are the representations of
the quiver Q, while positive representations of (A, S) are the representations of the
quiver Q satisfying the relations
(11.1) b⋆[ba] = [ba]a⋆ = a⋆b⋆ = 0.
In view of Corollary 10.14 and Proposition 10.15, the mutation µ2 establishes a
bijection between the set of right-equivalence classes of indecomposable positive rep-
resentations of (A, 0) different from the simple representation S2, and the same set for
(A, S). Since Q is a Dynkin quiver of type A3, by Gabriel’s theorem, an indecompos-
able positive representationM of (A, 0) is uniquely up to an isomorphism determined
by its dimension vector dimM = (dimM1, dimM2, dimM3), and these dimension
vectors are the positive roots of type A3 (note that in this case, the right-equivalence
classes are the same as isomorphism classes). Computing the images of these rep-
resentations under µ2, we obtain the correspondence between the dimension vectors
given in Table 1. We conclude that an indecomposable positive representation of
dimM (1,0,0) (0,0,1) (1,1,0) (0,1,1) (1,1,1)
dim µ2(M) (1,1,0) (0,1,1) (1,0,0) (0,0,1) (1,0,1)
Table 1. Indecomposable representations for A3 and the cyclic triangle.
QUIVERS WITH POTENTIALS I 51
(A, S) is uniquely up to right-equivalence determined by its dimension vector, and
these dimension vectors are given in the second line of Table 1, with the exception
of dimS2 = (0, 1, 0).
Example 11.2. Now let Q be the quiver with three vertices 1, 2, 3 and three arrows
a : 1 → 2, b : 2 → 3, and c : 1 → 3:
Again, the only QP on Q is (A, 0). We have µ2(A, 0) = (A, S), where A is the arrow
span of the quiver Q given by
and S = b⋆[ba]a⋆. Again, positive representations of (A, 0) are the representations of
the quiver Q, while positive representations of (A, S) are the representations of the
quiver Q satisfying the relations (11.1).
We consider indecomposable positive representations of (A, 0) with the dimension
vector (n, n, n) for some n ≥ 1. Assume that K is algebraically closed. Since Q is
an extended Dynkin quiver of type A
2 , and (n, n, n) is an isotropic imaginary root,
by Kac’s extension of Gabriel’s theorem, the isomorphism classes of indecomposable
Q-representations of this dimension form a 1-parametric family. An easy check shows
that these representations break into three right-equivalence classes. Their represen-
tatives can be described as follows. For each of them we have M1 =M2 =M3 = K
and two of the maps aM , bM , cM are equal to the identity map I, while the third one
is the nilpotent Jordan block N . If aM = N (resp. bM = N , cM = N) then we
denote the corresponding Q-representation by M(a) (resp. M(b), M(c)). In view of
(10.7), ifM is one of these representations then µ2(M) =M is positive, and we have
M 2 = coker bM ⊕ ker aM
(note that since S = 0, we have γ = 0). It follows that M(c) has dimension vector
(n, 0, n), with the maps [ba]
: Kn → KN given by [ba]
= I, c
= N .
Also both representations M(a) and M(b) have dimension vector (n, 1, n). In each
of them, the arrows [ba] and c act as [ba] = N, c = I. We also have b⋆
= 0, while
the map a⋆
: K → Kn has im a⋆
= kerN ; similarly, a⋆
= 0, while the map
: Kn → K has ker b⋆
= im N .
Example 11.3. Our last example deals with the QP (A, S) from Example 8.6.
Thus, the quiver in question has three vertices 1, 2, 3 and six arrows a1, a2 : 1 → 2,
52 HARM DERKSEN, JERZY WEYMAN, AND ANDREI ZELEVINSKY
b1, b2 : 2 → 3 and c1, c2 : 3 → 1; and the potential S is given by (8.2).
b2 ��=
To specify a positive representationM of (A, S), we need to define three vector spaces
M1,M2,M3, and six linear maps (a1)M , (a2)M :M1 → M2, (b1)M , (b2)M : M2 → M3,
and (c1)M , (c2)M :M3 → M1. In our case, J(S) is the closure of the ideal in R〈〈A〉〉
generated by six elements
c1b1, b1a1, a1c1, c2b2, b2a2, a2c2.
Thus, all the compositions (c1)M(b1)M , . . . , (a2)M(c2)M must be equal to 0.
We first consider the indecomposable positive representation M of (A, S) given
(11.2) M1 =M2 = K, M3 = 0; (a1)M = (a2)M = 1.
Let us compute µ2(M) = (M,V ). First of all, the QP µ2(A, S) = (A, S) was
computed in Example 8.6: recall that the arrows in A are a⋆1, a
2, [b1a2], [b2a1],
and the potential S is given by
S = [b1a2]a
1 + [b2a1]a
To compute M and V , we apply (10.6) and (10.7) to the triangle (10.5) given by
Min = K
2, Mk =M2 = K, Mout = {0}, α =
(so we have β = 0 and γ = 0). It follows that V = {0}, i.e., µ2(M) is positive; we
also have M 1 =M1 = K, M 3 =M3 = {0}, and
M 2 = kerα = K ·
(this is the third term in the decomposition of Mk in (10.7)). Since M3 = 0, the
arrows b⋆1, b
2, [b1a2], [b2a1] act as 0 in M . As for a
1 and a
2, their action is given by
the second equality in (10.10) (note that the choice of a splitting (10.9) is immaterial
here). Namely, identifying M 2 with K via choosing
as the standard basis
vector, we obtain
(a⋆1)M = 1, (a
2)M = −1
as maps M2 = K → K =M 1.
Note that the resulting representation µ2(M) can be conveniently described as
follows: by renumbering the vertices of our quiver via
(11.3) 1′ = 2, 2′ = 1, 3′ = 3,
and setting
(11.4) a′1 = −a
2 = a
1 = [b1a2], b
2 = [b2a1], c
1 = −b
2 = b
the representation µ2(M) gets identified with the initial representation M of the
initial QP (A, S).
QUIVERS WITH POTENTIALS I 53
The mutation µ1(M) can be computed in a similar way. But since we have already
computed the QP µ2(A, S) = (A, S), we find it more convenient to renumber the
vertices via 1′ = 3, 2′ = 1, 3′ = 2, so that µ1(M) gets identified with µ2(M
′), where
M ′ is given by:
(11.5) M ′1 = 0, M
3 = K, ; (b1)M ′ = (b2)M ′ = 1.
Now the triangle (10.5) is given by
M ′in = {0}, M
2 = K, M
out = K
2, β =
(so we have α = 0 and γ = 0). It follows that µ2(M
′) is positive, and we have
M ′1 =M
1 = {0}, M
3 = K, and
M ′2 = K
2/K ·
(this is the first term in the decomposition of Mk in (10.7)). Since M
1 = 0, the
arrows a⋆1, a
2, [b1a2], [b2a1] act as 0 in M
′. As for b⋆1 and b
2, their action is given by
the first equality in (10.10) (note that the choice of a splitting (10.8) is immaterial
here). Namely, identifying M ′2 with K via choosing π(
) = −π(
) as the
standard basis vector, we obtain
(b⋆1)M ′ = −1, (b
2)M ′ = 1
as maps M ′3 = K → K =M ′2.
As above, by renumbering the vertices of our quiver via
(11.6) 1′ = 1, 2′ = 3, 3′ = 2,
and setting
(11.7) a′1 = [b2a1], a
2 = [b1a2], b
1 = b
2 = −b
1 = a
2 = −a
the resulting representation µ2(M
′) gets identified with the initial representationM ′.
We now include the representations M and M ′ given by (11.2) and (11.5) into
a family of positive representations of (A, S) defined as follows: for every pair of
nonnegative integers (m,n) 6= (0, 0), we define the positive representation M =
M(m,n) of (A, S) by setting
(11.8) M1 = K
m, M2 = K
m+n, M3 = K
(a1)M =
, (a2)M =
, (b1)M =
,(11.9)
(b2)M =
, (c1)M = 0, (c2)M = 0,
where In is the n× n identity matrix.
We refer to the representations M(m,n) as well as those obtained from them by
renumbering the vertices as band representations ; they are a special case of band
modules studied in [11, 21] in the context of string algebras. Note that both repre-
sentations M and M ′ treated above are indeed special cases of band representations:
54 HARM DERKSEN, JERZY WEYMAN, AND ANDREI ZELEVINSKY
we have M = M(1, 0), M ′ = M(0, 1). By a direct generalization of the above
computations, we obtain the following proposition.
Proposition 11.4.
(1) If m ≥ n then after renumbering of vertices as in (11.3) and the change of
arrows as in (11.4), the representation µ2(M(m,n)) can be identified with
M(m− n, n).
(2) If m ≤ n then after renumbering of vertices as in (11.6) and the change of
arrows as in (11.7), the representation µ2(M(m,n)) can be identified with
M(m,n−m).
Remembering Theorem 10.13, we obtain the following corollary.
Corollary 11.5.
(1) After renumbering of vertices as in (11.3), the representation µ1(M(m,n))
becomes right-equivalent to M(m+ n, n).
(2) After renumbering of vertices as in (11.6), the representation µ3(M(m,n))
becomes right-equivalent to M(m,m+ n).
Corollary 11.6. The class of band representations is closed under mutations.
Note that if we iterate the mutations in Proposition 11.4, the pair (m,n) gets
transformed according to the Euclid algorithm for finding gcd(m,n). Thus, after a
sequence of mutations (and appropriate renumberings of vertices), every M(m,n)
can be transformed into M(gcd(m,n), 0). Since M(d, 0) is obviously isomorphic to
the direct sum of d copies of M(1, 0), by backtracking this sequence of mutations,
we obtain the following well-known corollary.
Corollary 11.7. The representation M(m,n) is indecomposable if and only if m
and n are relatively prime. Furthermore, if gcd(m,n) = d then M(m,n) is right-
equivalent to the direct sum of d copies of M(m/d, n/d).
Remark 11.8. By the same methods as above, one can compute all the mutations for
another family of representations of the QP (A, S) in Example 11.3: string modules
introduced and studied in [11, 21].
12. Some open problems
Here we collect some natural questions that we find important for better under-
standing of QPs and their representations. In what follows, suppose that (A, S) is a
reduced QP with the Jacobian algebra P(A, S). Let M(A, S) denote the category of
finite dimensional P(A, S)-modules. Suppose also that k ∈ Q0 is a vertex satisfying
(5.1), so that the mutated reduced QP µk(A, S) is well-defined.
Question 12.1. Is the isomorphism class of P(A, S) determined by the equivalence
class of the category M(A, S)?
Question 12.2. Is the isomorphism class of P(µk(A, S)) determined by the isomor-
phism class of P(A, S)?
Question 12.3. Is the category M(µk(A, S)) determined up to equivalence by
M(A, S)?
QUIVERS WITH POTENTIALS I 55
Note that the right-equivalence class of (A, S) is not determined by the isomor-
phism class of the Jacobian algebra P(A, S). In fact, we can construct a QP (A, S)
which is not right-equivalent to (A, cS) for some nonzero c ∈ K, while we obviously
have P(A, S) = P(A, cS) (the possibility of such an example was brought to our
attention by Bill Crawley-Boevey).
We conclude with the following intriguing question.
Question 12.4. Is there a proper analogue of the cluster category for a non-acyclic
quiver with potential?
13. Appendix. Proof of Lemma 4.12
We include Lemma 4.12 into a more general setup. We call a K-vector space V a
C-space (for the lack of a better term) if V has an increasing filtration {0} = V0 ⊆
V1 ⊆ · · · such that all Vn are finite dimensional, and V =
n≥0 Vn. (Equivalently, V
is either finite dimensional, or it has countable dimension.) The class of C-spaces is
clearly closed under taking subspaces, quotient spaces, finite direct sums, and finite
tensor products. We always consider C-spaces equipped with discrete topology; in
particular, this applies to the base field K.
We refer to the dual space V ⋆ of a C-space V as a D-space (the dual is understood
as the space of all linear forms V → K). Most of the properties ofD-spaces discussed
below are undoubtedly well-known; for the convenience of the reader, we provide a
self-contained treatment.
Example 13.1. The complete path algebra R〈〈A〉〉 can be naturally viewed as a
D-space V ⋆, corresponding to the C-space V = ⊕∞d=0(A
, and the filtration (Vn)
given by
Vn = ⊕
d=0(A
(n ≥ 1).
For a subspace W of V , we denote by W⊥ ⊂ V ⋆ its orthogonal complement, that
W⊥ = {f ∈ V ⋆ | f(W ) = 0}.
We make V ⋆ into a topological vector space by taking the sets V ⊥n for all n ≥ 0
as a basic system of open neighborhoods of 0. In particular, in Example 13.1, we
have V ⊥n = m(A)
n, so the D-space topology on R〈〈A〉〉 coincides with the topology
introduced in Section 2.
Since every v ∈ V belongs to some Vn, a sequence f1, f2, . . . converges in V
⋆ if and
only if, for every v ∈ V , the sequence (fk(v)) stabilizes as k → ∞. This implies in
particular that W⊥ is a closed subspace of V ⋆ for every subspace W of V . In fact,
the converse is also true.
Lemma 13.2. A vector subspace Z of V ⋆ is closed if and only if Z =W⊥ for some
subspace W of V .
Proof. Let Z be a vector subspace of V ⋆. Let
W = {v ∈ V | f(v) = 0 for f ∈ Z}.
It suffices to show that W⊥ is contained in the closure Z of Z. Let f ∈ W⊥.
Restricting f to each finite-dimensional subspace Vn of V , we conclude that f |Vn=
56 HARM DERKSEN, JERZY WEYMAN, AND ANDREI ZELEVINSKY
hn |Vn for some hn ∈ Z. Thus, the sequence h1, h2, . . . , converges to f , implying that
f ∈ Z, as required. �
In view of Lemma 13.2, for every closed subspace Z of V ⋆, the spaces Z and V ⋆/Z
can be naturally viewed as D-spaces: indeed, we have
Z =W⊥ = (V/W )⋆, V ⋆/Z = V ⋆/W⊥ = W ⋆
for some subspace W of V . The following lemma is immediate from the definitions.
Lemma 13.3. For every closed subspace Z ⊆ V ⋆, the D-space topologies on Z and
V ⋆/Z coincide with the topologies induced from V ⋆. In particular, the embedding
Z → V ⋆ and the projection V ⋆ → V ⋆/Z are continuous.
Lemma 13.4. If Z1 and Z2 are closed subspaces of V
⋆, then Z1 + Z2 is a closed
subspace of V ⋆ as well.
Proof. By Lemma 13.2, Z1 = W
1 and Z2 = W
2 for some subspaces W1 and W2
of V . Choosing some direct complements of W1 ∩W2 in W1 and W2, and a direct
complement of W1 +W2 in V , it is easy to see that
Z1 + Z2 = W
2 = (W1 ∩W2)
proving that Z1 + Z2 is closed. �
Lemma 13.5. Let U and V be C-spaces, and U⋆ and V ⋆ be the corresponding D-
spaces. A linear map α : U⋆ → V ⋆ is continuous if and only if α = β⋆ for some
linear map β : V → U .
Proof. First let us show that α = β⋆ is continuous. By the definition, it is enough to
show that, for every n, there exists an index k such that U⊥k ⊂ α
−1(V ⊥n ). Since the
subspace β(Vn) ⊂ U is finite dimensional, it is contained in some Uk, implying the
desired inclusion U⊥k ⊂ α
−1(V ⊥n ).
Conversely, suppose α : U⋆ → V ⋆ is a continuous linear map. Let v ∈ V . Then
the linear form f 7→ α(f)(v) is a continuous linear map U⋆ → K, and so its kernel is
a closed subspace of U⋆. Using Lemma 13.2, we conclude that there exists a unique
u ∈ U such that α(f)(v) = f(u) for all f ∈ U⋆. The correspondence v 7→ u is the
desired linear map β : V → U such that α = β⋆. �
Lemma 13.6. Any continuous linear map of D-spaces α : U⋆ → V ⋆ sends closed
vector subspaces of U⋆ to closed vector subspaces of V ⋆.
Proof. Let Z ⊆ U⋆ be a closed vector subspace. By Lemma 13.2, Z =W⊥ for some
vector subspace W ⊂ U . Also by Lemma 13.5, we have α = β⋆ for a linear map
β : V → U . The definitions imply that α(Z) = β⋆(W⊥) = (β−1(W ))⊥, hence α(Z)
is a closed subspace of V ⋆, as claimed. �
We will call a D-space V ⋆ a D-algebra if it has a structure of an associative
K-algebra such that V ⊥m V
n ⊂ V
m+n for all m,n ≥ 0. In particular, R〈〈A〉〉 is a
D-algebra.
Lemma 13.7. If I1, . . . , IN are closed subspaces in a D-algebra V
⋆, then the subspace
I1f1+ · · ·+INfN is closed for every f1, . . . , fN ∈ V
⋆. In particular, finitely generated
left ideals in V ⋆ are closed.
QUIVERS WITH POTENTIALS I 57
Proof. By the definition of a D-algebra, the operator of right multiplication with any
f ∈ V ⋆ is continuous. Thus each subspace Ikfk is closed by Lemma 13.6, and our
assertion follows from Lemma 13.4. �
Recall from Definition 3.4, that the trace space of a D-algebra V ⋆ is the quo-
tient Tr(V ⋆) = V ⋆/{V ⋆, V ⋆}, where {V ⋆, V ⋆} is the closure of the vector subspace
in V ⋆ spanned by all commutators. We denote by π : V ⋆ → Tr(V ⋆) the canonical
projection. By Lemma 13.3, π is continuous with respect to the D-space topologies.
In view of Proposition 3.5, the assertion of Lemma 4.12 is a special case of the
following.
Lemma 13.8. Let I be a closed (two-sided) ideal of a D-algebra V ⋆, and J be
the closure of an ideal generated by finitely many elements f1, f2, . . . , fN . Then the
subspace π(IJ) ⊆ Tr(V ⋆) is equal to π(If1 + · · ·+ IfN).
Proof. Let J0 be the ideal generated by f1, f2, . . . , fN , that is, the linear span of
elements of the form ufkv with u, v ∈ V
⋆ and k = 1, . . . , N . Thus the ideal IJ0
is the linear span of elements of the form gufkv with g ∈ I. By the definition, we
have π(gufkv) = π(vgufk), and so π(IJ
0) = π(If1 + · · ·+ IfN). Since IJ
0 is dense
in IJ , it follows that π(IJ0) is dense in π(IJ). On the other hand, the subspace
π(If1 + · · ·+ IfN) ⊆ Tr(V
⋆) is closed by Lemmas 13.7 and 13.6. We conclude that
π(IJ0) = π(If1 + · · ·+ IfN) = π(IJ), as required. �
Acknowledgments
We thank Victor Ginzburg for helpful comments and bibliographic guidance, Bill
Crawley-Boevey for valuable remarks, Daniel Labardini Fragoso for careful reading
of the manuscript and many useful suggestions and Christopher Herzog for making
us aware of and discussing the use of quivers with potentials in theoretical physics.
After this paper was completed, we have been informed by Maxim Kontsevich that
he has rediscovered some of the results in Sections 6 and 7 in the context of A∞-
categories.
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Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA
E-mail address : hderksen@umich.edu
Department of Mathematics, Northeastern University, Boston, MA 02115
E-mail address : j.weyman@neu.edu
Department of Mathematics, Northeastern University, Boston, MA 02115
E-mail address : andrei@neu.edu
	1. Introduction
	2. Quivers and path algebras
	3. Potentials and their Jacobian ideals
	4. Quivers with potentials
	5. Mutations of quivers with potentials
	6. Some mutation invariants
	7. Nondegenerate QPs
	8. Rigid QPs
	9. Relation to cluster-tilted algebras
	10. Decorated representations and their mutations
	11. Some three-vertex examples
	12. Some open problems
	13. Appendix. Proof of Lemma 4.12
	Acknowledgments
	References
