A separable deformation of the quaternion group algebra
A SEPARABLE DEFORMATION OF THE QUATERNION
GROUP ALGEBRA
NURIT BARNEA AND YUVAL GINOSAR
Abstract. The Donald-Flanigan conjecture asserts that for any finite group
G and any field k, the group algebra kG can be deformed to a separable algebra.
The minimal unsolved instance, namely the quaternion group Q8 over a field
k of characteristic 2 was considered as a counterexample. We present here a
separable deformation of kQ8. In a sense, the conjecture for any finite group
is open again.
1. Introduction
In their paper [1], J.D. Donald and F.J. Flanigan conjectured that any group
algebra kG of a finite group G over a field k can be deformed to a semisimple
algebra even in the modular case, namely where the order of G is not invertible
in k. A more customary formulation of the Donald-Flanigan (DF) conjecture is
by demanding that the deformed algebra [kG]t should be separable, i.e. it remains
semisimple when tensored with the algebraic closure of its base field. If, additionally,
the dimensions of the simple components of [kG]t are in one-to-one correspondence
with those of the complex group algebra CG, then [kG]t is called a strong solution
to the problem.
The DF conjecture was solved for groups G which have either a cyclic p-Sylow
subgroup over an algebraically closed field [11] or a normal abelian p-Sylow sub-
group [5] where p =char(k), and for all but six reflection groups in any characteristic
[6, 7, 10]. In [4], it is claimed that the group algebra kQ8, where
Q8 = 〈σ, τ |σ
4 = 1, τσ = σ3τ, σ2 = τ2 〉
is the quaternion group of order 8 and k a field of characteristic 2, does not admit
a separable deformation. This result allegedly gives a counterexample to the DF
conjecture. However, as observed by M. Schaps, the proof apparently contains an
error (see §7).
The aim of this note is to present a separable deformation of kQ8, where k is
any field of characteristic 2, reopening the DF conjecture.
2. Preliminaries
Let k[[t]] be the ring of formal power series over k, and let k((t)) be its field of
fractions. Recall that the deformed algebra [kG]t has the same underlying k((t))-
vector space as k((t))⊗k kG, with multiplication defined on basis elements
(2.1) g1 ∗ g2 := g1g2 +
Ψi(g1, g2)t
i, g1, g2 ∈ G
Date: November 13, 2018.
http://arxiv.org/abs/0704.1556v1
2 NURIT BARNEA AND YUVAL GINOSAR
and extended k((t))-linearly (such that t is central). Here g1g2 is the group multi-
plication. The functions Ψi : G×G → kG satisfy certain cohomological conditions
induced by the associativity of [kG]t [3, §1 ; §2].
Note that the set of equations (2.1) determines a multiplication on the free
k[[t]]-module Λt spanned by the elements {g}g∈G such that kG ≃ Λt/〈tΛt〉 and
[kG]t ≃ Λt ⊗k[[t]] k((t)). In a more general context, namely over a domain R which
is not necessarily local, the R-module Λt which determines the deformation, is
required only to be flat rather than free [2, §1].
In what follows, we shall define the deformed algebra [kG]t by using generators
and relations. These will implicitly determine the set of equations (2.1).
3. Sketch of the construction
Consider the extension
(3.1) [β] : 1 → C4 → Q8 → C2 → 1,
where C2 = 〈 τ̄ 〉 acts on C4 = 〈σ 〉 by
η : C2 → Aut(C4)
η(τ̄ ) : σ 7→ σ3(= σ−1),
and the associated 2-cocycle β : C2 × C2 → C4 is given by
β(1, 1) = β(1, τ̄) = β(τ̄ , 1) = 1, β(τ̄ , τ̄) = σ2.
The group algebra kQ8 (k any field) is isomorphic to the quotient kC4[y; η]/〈 q(y) 〉,
where kC4[y; η] is a skew polynomial ring [9, §1.2], whose indeterminate y acts
on the ring of coefficients kC4 via the automorphism η(τ̄ ) (extended linearly) and
where
(3.2) q(y) := y2 − σ2 ∈ kC4[y; η]
is central. The above isomorphism is established by identifying τ with the indeter-
minate y.
Suppose now that Char(k) = 2. The deformed algebra [kQ8]t is constructed as
follows.
In §4.1 the subgroup algebra kC4 is deformed to a separable algebra [kC4]t which
is isomorphic to K⊕k((t))⊕k((t)), where K is a separable field extension of k((t))
of degree 2.
The next step (§4.2) is to construct an automorphism ηt of [kC4]t which agrees
with the action of C2 on kC4 when specializing t = 0. This action fixes all three
primitive idempotents of [kC4]t. By that we obtain the skew polynomial ring
[kC4]t[y; ηt].
In §5 we deform q(y) = y2 + σ2 to qt(y), a separable polynomial of degree 2 in
the center of [kC4]t[y; ηt].
By factoring out the two-sided ideal generated by qt(y), we establish the defor-
mation
[kQ8]t := [kC4]t[y; ηt]/〈 qt(y) 〉.
In §6 we show that [kQ8]t as above is separable. Moreover, passing to the
algebraic closure k((t)) we have
[kQ8]t ⊗k((t)) k((t)) ≃
k((t))⊕M2(k((t))).
A SEPARABLE DEFORMATION OF THE QUATERNION GROUP ALGEBRA 3
This is a strong solution to the DF conjecture since its decomposition to simple
components is the same as
CQ8 ≃
C⊕M2(C).
4. A Deformation of kC4[y; η]
4.1. We begin by constructing [kC4]t, C4 = 〈σ 〉. Recall that
kC4 ≃ k[x]/〈x
4 + 1 〉
by identifying σ with x+ 〈x4+1 〉. We deform the polynomial x4+1 to a separable
polynomial pt(x) as follows.
Let k[[t]]∗ be the group of invertible elements of k[[t]] and denote by
U := {1 + zt|z ∈ k[[t]]∗}
its subgroup of 1-units (when k = F2, U is equal to k[[t]]
a ∈ k[[t]] \ k[[t]]∗
be a non-zero element, and let
b, c, d ∈ U, (c 6= d),
such that
π(x) := x2 + ax+ b
is an irreducible (separable) polynomial in k((t))[x]. Let
pt(x) := π(x)(x + c)(x + d) ∈ k((t))[x].
Then the quotient k((t))[x]/〈 pt(x) 〉 is isomorphic to the direct sum K ⊕ k((t)) ⊕
k((t)), where K := k((t))[x]/〈π(x) 〉. The field extension K/k((t)) is separable and
of dimension 2.
Note that pt=0(x) = x
4+1 and that only lower order terms of the polynomial were
deformed. Hence, the quotient k[[t]][x]/〈 pt(x) 〉 is k[[t]]-free and k((t))[x]/〈 pt(x) 〉
indeed defines a deformation [kC4]t of kC4 ≃ k[x]/〈x
4+1 〉. The new multiplication
σi∗σj of basis elements (2.1) is determined by identifying σi with x̄i := xi+〈 pt(x) 〉.
We shall continue to use the term x̄ in [kC4]t rather than σ.
Assume further that there exists w ∈ k[[t]] such that
(4.1) (x + w)(x + c)(x+ d) = xπ(x) + a
(see example 4.3). Then K ≃ ([kC4]t)e1, where
(4.2) e1 =
(x̄+ w)(x̄ + c)(x̄ + d)
The two other primitive idempotents of [kC4]t are
(4.3) e2 =
c(x̄ + d)π(x̄)
a(c+ d)
, e3 =
d(x̄ + c)π(x̄)
a(c+ d)
4 NURIT BARNEA AND YUVAL GINOSAR
4.2. Let
ηt : k((t))[x] → k((t))[x]
be an algebra endomorphism determined by its value on the generator x as follows.
(4.4) ηt(x) := xπ(x) + x+ a.
We compute ηt(π(x)), ηt(x+ c) and ηt(x+ d):
ηt(π(x)) = ηt(x)
2 + aηt(x) + b = x
2π(x)2 + x2 + a2 + axπ(x) + ax+ a2 + b
= π(x)(x2π(x) + ax+ 1).
By (4.1),
(4.5) ηt(π(x)) = π(x) + x(x+ w)pt(x) ∈ 〈π(x) 〉.
Next,
ηt(x+ c) = xπ(x) + x+ a+ c.
By (4.1),
(4.6) ηt(x+ c) = (x+ c)[(x + w)(x + d) + 1] ∈ 〈x+ c 〉.
Similarly,
(4.7) ηt(x+ d) = (x+ d)[(x + w)(x + c) + 1] ∈ 〈x+ d 〉.
By (4.5), (4.6) and (4.7), we obtain that ηt(pt(x)) ∈ 〈 pt(x) 〉, and hence ηt induces
an endomorphism of k((t))[x]/〈 pt(x) 〉 which we continue to denote by ηt. As can
easily be verified, the primitive idempotents given in (4.2) and (4.3) are fixed under
(4.8) ηt(ei) = ei, i = 1, 2, 3,
whereas
(4.9) ηt(x̄e1) = ηt(x̄)e1 = (x̄π(x̄) + x̄+ a)e1 = (x̄+ a)e1.
Hence, ηt induces an automorphism of K of order 2 while fixing the two copies of
k((t)) pointwise. Furthermore, one can easily verify that
ηt=0(x̄) = x̄
Consequently, the automorphism ηt of [kC4]t agrees with the automorphism η(τ̄ )
of kC4 when t = 0. The skew polynomial ring
[kC4]t[y; ηt] = (k((t))[x]/〈 pt(x) 〉)[y; ηt]
is therefore a deformation of kC4[y; η].
Note that by (4.8), the idempotents ei, i = 1, 2, 3 are central in [kC4]t[y; ηt] and
hence
(4.10) [kC4]t[y; ηt] =
[kC4]t[y; ηt]ei.
A SEPARABLE DEFORMATION OF THE QUATERNION GROUP ALGEBRA 5
4.3. Example. The following is an example for the above construction.
t+ t2 + t3
1 + t
, b := 1 + t2 + t3, c :=
1 + t
, d := 1 + t+ t2, w := t.
These elements satisfy equation (4.1):
(x+ w)(x+ c)(x + d) = (x+ t)(x +
1 + t
)(x+ 1 + t+ t2)
= x3 +
t+ t2 + t3
1 + t
x2 + (1 + t2 + t3)x +
t+ t2 + t3
1 + t
= xπ(x) + a.
The polynomial
π(x) = x2 +
t+ t2 + t3
1 + t
x+ 1 + t2 + t3
does not admit roots in k[[t]]/〈 t2 〉, thus it is irreducible over k((t)).
5. A Deformation of q(y)
The construction of [kQ8]t will be completed once the product τ̄ ∗ τ̄ is defined.
For this purpose the polynomial q(y) (3.2), which determined the ordinary multi-
plication τ2, will now be developed in powers of t.
For any non-zero element z ∈ k[[t]] \ k[[t]]∗, let
(5.1) qt(y) := y
2 + zx̄π(x̄)y + x̄2 + ax̄ ∈ [kC4]t[y; ηt].
Decomposition of (5.1) with respect to the idempotents e1, e2, e3 yields
(5.2) qt(y) = (y
2 + b)e1 + [y
2 + zay + c(c+ a)]e2 + [y
2 + zay + d(d+ a)]e3.
We now show that qt(y) is in the center of [kC4]t[y; ηt] :
First, the leading term y2 is central since the automorphism ηt is of order 2.
Next, by (4.8), the free term be1 + c(c + a)e2 + d(d + a)e3 is invariant under the
action of ηt and hence central. It is left to check that the term za(e2 + e3)y is
central. Indeed, since e2 and e3 are ηt-invariant, then za(e2 + e3)y commutes both
with [kC4]t[y; ηt]e2 and [kC4]t[y; ηt]e3. Furthermore, by orthogonality
za(e2 + e3)y · [kC4]t[y; ηt]e1 = [kC4]t[y; ηt]e1 · za(e2 + e3)y = 0,
and hence za(e2 + e3)y commutes with [kC4]t[y; ηt].
Consequently, 〈 qt(y) 〉 = qt(y)[kC4]t[y; ηt] is a two-sided ideal.
Now, as can easily be deduced from (5.1),
(5.3) qt=0(y) = y
2 + x̄2 = q(y),
where the leading term y2 remains unchanged. Then
[kQ8]t := [kC4]t[y; ηt]/〈 qt(y) 〉
is a deformation of kQ8, identifying τ̄ with ȳ := y + 〈 qt(y) 〉.
6 NURIT BARNEA AND YUVAL GINOSAR
6. Separability of [kQ8]t
Finally, we need to prove that the deformed algebra [kQ8]t is separable. More-
over, we prove that its decomposition to simple components over the algebraic
closure of k((t)) resembles that of CQ8. By (4.10), we obtain
(6.1) [kQ8]t =
[kC4]t[y; ηt]ei/〈 qt(y)ei 〉.
We handle the three summands in (6.1) separately:
By (5.2),
[kC4]t[y; ηt]e1/〈 qt(y)e1 〉 ≃ K[y; ηt]/〈 y
2 + b 〉 ≃ Kf ∗ C2.
The rightmost term is the crossed product of the group C2 := 〈 τ̄ 〉 acting faithfully
on the field K = [kC4]te1 via ηt (4.9), with a twisting determined by the 2-cocycle
f : C2 × C2 → K
f(1, 1) = f(1, τ̄) = f(τ̄ , 1) = 1, f(τ̄ , τ̄ ) = b.
This is a central simple algebra over the subfield of invariants k((t)) [8, Theorem
4.4.1]. Evidently, this simple algebra is split by k((t)), i.e.
(6.2) [kC4]t[y; ηt]e1/〈 qt(y)e1 〉 ⊗k((t)) k((t)) ≃ M2(k((t))).
Next, since ηt is trivial on [kC4]te2, the skew polynomial ring [kC4]te2[y; ηt] is
actually an ordinary polynomial ring k((t))[y]. Again by (5.2),
[kC4]t[y; ηt]e2/〈 qt(y)e2 〉 ≃ k((t))[y]/〈 y
2 + zay + c(c+ a) 〉.
Similarly,
[kC4]t[y; ηt]e3/〈 qt(y)e3 〉 ≃ k((t))[y]/〈 y
2 + zay + d(d+ a) 〉.
The polynomials y2 + zay + c(c+ a) and y2 + zay + d(d + a) are separable (since
za is non-zero). Thus, both [kC4]t[y; ηt]e2/〈 qt(y)e2 〉 and [kC4]t[y; ηt]e3/〈 qt(y)e3 〉
are separable k((t))-algebras, and for i = 2, 3
(6.3) [kC4]t[y; ηt]ei/〈 qt(y)ei 〉 ⊗k((t)) k((t)) ≃ k((t)) ⊕ k((t)).
Equations (6.1), (6.2) and (6.3) yield
[kQ8]t ⊗k((t)) k((t)) ≃
k((t)) ⊕M2(k((t)))
as required.
7. Acknowledgement
We wish to thank M. Schaps for pointing out to us that there is an error in the
attempted proof in [4] that the quaternion group is a counterexample to the DF
conjecture. Here is her explanation: The given relations for the group algebra are
incorrect. Using the notation in pages 166-7 of [4], if a = 1 + i, b = 1 + j and
z = i2 = j2, then ab + ba = ij(1 + z) while a2 = b2 = 1 + z. There is a further
error later on when the matrix algebra is deformed to four copies of the field, since
a non-commutative algebra can never have a flat deformation to a commutative
algebra.
A SEPARABLE DEFORMATION OF THE QUATERNION GROUP ALGEBRA 7
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for Sn, Trans. Amer. Math. Soc. 349 (1997), no. 8, 3353–3371.
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Department of Mathematics, University of Haifa, Haifa 31905, Israel
E-mail address: ginosar@math.haifa.ac.il
	1. Introduction
	2. Preliminaries
	3. Sketch of the construction
	4. A Deformation of kC4[y;]
	4.1. 
	4.2. 
	4.3. Example
	5. A Deformation of q(y)
	6. Separability of [kQ8]t
	7. Acknowledgement
	References
