Quantum Group of Isometries in Classical and Noncommutative Geometry
arXiv:0704.0041v4  [math.QA]  26 Oct 2007
Quantum Group of Isometries in Classical and
Noncommutative Geometry
Debashish Goswami 1
Stat-Math Unit, Kolkata Centre,
Indian Statistical Institute
203, B. T. Road, Kolkata 700 108, India
Abstract
We formulate a quantum generalization of the notion of the group of
Riemannian isometries for a compact Riemannian manifold, by intro-
ducing a natural notion of smooth and isometric action by a compact
quantum group on a classical or noncommutative manifold described
by spectral triples, and then proving the existence of a universal object
(called the quantum isometry group) in the category of compact quan-
tum groups acting smoothly and isometrically on a given (possibly
noncommutative) manifold satisfying certain regularity assumptions.
In fact, we identify the quantum isometry group with the universal ob-
ject in a bigger category, namely the category of ‘quantum families of
smooth isometries’, defined along the line of Woronowicz and Soltan.
We also construct a spectral triple on the Hilbert space of forms on a
noncommutative manifold which is equivariant with respect to a nat-
ural unitary representation of the quantum isometry group. We give
explicit description of quantum isometry groups of commutative and
noncommutative tori, and in this context, obtain the quantum double
torus defined in [11] as the universal quantum group of holomorphic
isometries of the noncommutative torus.
1 Introduction
Since the formulation of quantum automorphism groups by Wang ([15],
[16]), following suggestions of Alain Connes, many interesting examples of
such quantum groups, particularly the quantum permutation groups of finite
sets and finite graphs, have been extensively studied by a number of mathe-
maticians (see, e.g. [1], [2], [17] and references therein), who have also found
applications to and interaction with areas like free probability and subfactor
theory. The underlying basic principle of defining a quantum automorphism
group corresponding to some given mathematical structure (for example, a
1The author gratefully acknowldges support obtained from the Indian National
Academy of Sciences through the grants for a project on ‘Noncommutative Geometry
and Quantum Groups’, and also wishes to thank The Abdus Salam ICTP (Trieste), where
a major part of the work was done during a visit as Junior Assciate.
http://arxiv.org/abs/0704.0041v4
finite set, a graph, a C∗ or von Neumann algebra) consists of two steps :
first, to identify (if possible) the group of automorphisms of the structure as
a universal object in a suitable category, and then, try to look for the univer-
sal object in a similar but bigger category by replacing groups by quantum
groups of appropriate type. However, most of the work done so far concern
some kind of quantum automorphism groups of a ‘finite’ structure, for ex-
ample, of finite sets or finite dimensional matrix algebras. It is thus quite
natural to try to extend these ideas to the ‘infinite’ or ‘continuous’ mathe-
matical structures, for example classical and noncommutative manifolds. In
the present article, we have made an attempt to formulate and study the
quantum analogues of the groups of Riemannian isometries, which play a
very important role in the classical differential geometry. The group of Rie-
mannian isometries of a compact Riemannian manifold M can be viewed as
the universal object in the category of all compact metrizable groups acting
on M , with smooth and isometric action. Therefore, to define the quantum
isometry group, it is reasonable to consider a category of compact quantum
groups which act on the manifold (or more generally, on a noncommutative
manifold given by spectral triple) in a ‘nice’ way, preserving the Riemannian
structure in some suitable sense, to be precisely formulated. In this article,
we have given a definition of such ‘smooth and isometric’ action by a com-
pact quantum group on a (possibly noncommutative) manifold, extending
the notion of smooth and isometric action by a group on a classical mani-
fold. Indeed, the meaning of isometric action is nothing but that the action
should commute with the ‘Laplacian’ coming from the spectral triple, and
we should mention that this idea was already present in [2], though only in
the context of a finite metric space or a finite graph. The universal object
in the category of such quantum groups, if it exists, should be thought of
as the quantum analogue of the group of isometries, and we have been able
to prove its existence under some regularity assumptions, all of which can
be verified for a general compact connected Riemannian manifold as well
as the standard examples of noncommutative manifolds. Motivated by the
ideas of Woronowicz and Soltan, we actually consider a bigger category. The
isometry group of a classical manifold, viewed as a compact metrizable space
(forgetiing the group structure), can be seen to be the universal object of a
category whose object-class consists of subsets (not necessarily subgroups)
of the set of smooth isometries of the manifold. Then it can be proved
that this universal compact set has a canonical group structure. A natural
quantum analogue of this has been formulated by us, called the category of
‘quantum families of smooth isometries’. The underlying C∗-algebra of the
quantum isometry group has been identified with its universal object and
moreover, it is shown to be equipped with a canonical coproduct making it
into a compact quantum group.
We believe that a detailed study of quantum isometry groups will not
only give many new and interesting examples of compact quantum groups,
it will also contribute to the understanding of quantum group covariant
spectral triples. In fact, we have made some progress in this direction already
by constructing a spectral triple (which is often closely related to the original
spectral triple) on the Hilbert space of forms which is equivriant with respect
to a canonical unitary representation of the quantum isometry group.
In a companion article [3] with J. Bhowmick, we provide explicit compu-
tations of quantum isometry groups of a few classical and noncommutative
manifolds. However, we briefly quote some of main results of [3] in the
present article. One interesting observation is that the quantum isometry
group of the noncommutative two-torus Aθ (with the canonical spectral
triple) is (as a C∗ algebra) a direct sum of two commutative and two non-
commutative tori, and contains as a quantum subgroup (which is univer-
sal for certain class of isometric actions called holomorphic isometries) the
‘quantum double-torus’ discovered and studied by Hajac and Masuda ([11]).
2 Definition of the quantum isometry group
2.1 Isometry groups of classical manifolds
We begin with a well-known characterization of the isometry group of a (clas-
sical) compact Riemannian manifold. Let (M,g) be a compact Riemannian
manifold and let Ω1 = Ω1(M) be the space of smooth one-forms, which has
a right Hilbert-C∞(M)-module structure given by the C∞(M)-valued inner
product << ·, · >> defined by
<< ω, η >> (m) =< ω(m), η(m) > |m,
where < ·, · > |m is the Riemannian metric on the cotangent space T
mM at
the pointm ∈M . The Riemannian volume form allows us to make Ω1 a pre-
Hilbert space, and we denote its completion by H1. Let H0 = L
2(M,dvol)
and consider the de-Rham differential d as an unbounded linear map from
H0 toH1, with the natural domain C
∞(M) ⊂ H0, and also denote its closure
by d. Let L := −d∗d. The following identity can be verified by direct and
easy computation using the local coordinates :
(∂L)(φ,ψ) ≡ L(φ̄ψ)−L(φ̄)ψ−φ̄L(ψ) = 2 << dφ, dψ >> for φ,ψ ∈ C∞(M) (∗).
Proposition 2.1 A smooth map γ : M → M is a Riemannian isometry if
and only if γ commutes with L in the sense that L(f ◦ γ) = (L(f)) ◦ γ for
all f ∈ C∞(M).
Proof :
If γ commutes with L then from the identity (*) we get for m ∈ M and
φ,ψ ∈ C∞(M) :
< dφ|γ(m), dψ|γ(m) > |γ(m)
= << dφ, dψ >> (γ(m))
(∂L(φ,ψ) ◦ γ)(m)
∂L(φ ◦ γ, ψ ◦ γ)(m)
= << d(φ ◦ γ), d(ψ ◦ γ) >> (m)
= < d(φ ◦ γ)|m, d(ψ ◦ γ)|m > |m
= < (dγ|m)
∗(dφ|γ(m)), (dγ|m)
∗(dψ|γ(m)) > |m,
which proves that (dγ|m)
∗ : T ∗
M → T ∗mM is an isometry. Thus, γ is a
Riemannian isometry.
Conversely, if γ is an isometry, both the maps induced by γ on H0 and
H1, i.e. U
γ : H0 → H0 given by U
γ (f) = f ◦ γ and U
γ : H
1 → H1 given
by U1γ (fdφ) = (f ◦ γ)d(φ ◦ γ) are unitaries. Moreover, d ◦ U
γ = U
γ ◦ d on
C∞(M) ⊂ H0. From this, it follows that L = −d
∗d commutes with U0γ . ✷
Now let us consider a compact metrizable (i.e. second countable) space
Y with a continuous map θ : M × Y → M . We abbreviate θ(m, y) as ym
and denote by ξy the map M ∋ m 7→ ym. Let α : C(M) → C(M)⊗C(Y ) ∼=
C(M × Y ) be the map given by α(f)(m, y) := f(ym) for y ∈ Y , m ∈ M
and f ∈ C(M). For a state φ on C(Y ), denote by αφ the map (id⊗ φ) ◦ α :
C(M) → C(M). We shall also denote by C the subspace of C(M) ⊗ C(Y )
generated by elements of the form α(f)(1⊗ψ), f ∈ C(M), ψ ∈ C(Y ). Since
C(M) and C(Y ) are commutative algebras, it is easy to see that C is a
∗-subalgebra of C(M)⊗ C(Y ). Then we have the following
Theorem 2.2 (i) C is norm-dense in C(M)⊗C(Y ) if and only if for every
y ∈ Y , ξy is one-to-one.
(ii) The map ξy is C
∞ for every y ∈ Y if and only if αφ(C
∞(M)) ⊆ C∞(M)
for all φ.
(iii) Under the hypothesis of (ii), each ξy is also an isometry if and only if
αφ commutes with (L − λ)
−1 for all state φ and all λ in the resolvent of L
(equivalently, αφ commutes with the Laplacian L on C
∞(M)).
Proof :
(i) First, assume that ξy is one-to-one for all y. By Stone-Weirstrass Theo-
rem, it is enough to show that C separates points. Take (m1, y1) 6= (m2, y2)
in M × Y . If y1 6= y2, we can choose ψ ∈ C(Y ) which separates y1 and y2,
hence (1 ⊗ ψ) ∈ C separates (m1, y1) and (m2, y2). So, we can consider the
case when y1 = y2 = y (say), but m1 6= m2. By injectivity of ξy, we have
ym1 6= ym2, so there exists f ∈ C(M) such that f(ym1) 6= f(ym2), i.e.
α(f)(m1, y) 6= α(f)(m2, y). This proves the density of C.
For the converse, we argue as in the proof of Proposition 3.3 of [14].
Assume that C is dense in C(M)⊗ C(Y ), and let y ∈ Y , m1,m2 ∈ M such
that ym1 = ym2. That is, α(f)(1 ⊗ ψ)(m1, y) = α(f)(1 ⊗ ψ)(m2, y) for all
f ∈ C(M), ψ ∈ C(Y ). By the density of C we get χ(m1, y) = χ(m2, y) for
all χ ∈ C(M × Y ), so (m1, y) = (m2, y), i.e. m1 = m2.
(ii) The ‘if part’ of (ii) follows by considering the states corresponding to
point evaluation, i.e. C(Y ) ∋ ψ 7→ ψ(y), y ∈ Y . For the converse, we note
that an arbitrary state φ corresponds to a regular Borel measure µ on Y
so that φ(h) =
hdµ, and thus, αφ(f)(m) =
f(ym)dµ(y) for f ∈ C(M).
From this, by interchanging differentiation and integation (which is allowed
by the Dominated Convergence Theorem, since µ is a finite measure) we can
prove that αφ(f) is C
∞ whenever f is so.
The assertion (iii) follows from Proposition 2.1 in a straghtforward way.
Let us recall a few well-known facts about the Laplacian L, viewed as a
negative self-adjoint operator on the Hilbert space L2(M,dvol). It is known
(see [12] and references therein) that L has compact resolvents and all its
eigenvectors belong to C∞(M). Moreover, it follows from the Sobolev Em-
bedding Theorem that
Dom(Ln) = C∞(M).
Let {eij , j = 1, ..., di; i = 1, 2, ...} be the set of (normalised) eigenvectors of
L, where eij ∈ C
∞(M) is an eigenvector corresponding to the eigenvalue λi,
|λ1| < |λ2| < .... We have the following:
Lemma 2.3 The complex linear span of {eij} is norm-dense in C(M).
Proof :
This is a consequence of the asymptotic estimates of eigenvalues λi, as
well as the uniform bound of the eigenfunctions eij . For example, it is
known ([9],Theorem 1.2) that there exist constants C,C ′ such that ‖eij‖∞ ≤
C|λi|
4 , di ≤ C
′|λi|
2 , where n is the dimension of the manifoldM . Now,
for f ∈ C∞(M) ⊆
k≥1Dom(L
k), we write f as an a-priori L2-convergent
series
ij fijeij (fij ∈ C), and observe that
|fij |
2|λi|
2k < ∞ for every
k ≥ 1. Choose and fix sufficiently large k such that
i≥0 |λi|
n−1−2k < ∞,
which is possible due to the well-known Weyl asymptotics of eigenvalues of
L. Now, by the Cauchy-Schwarz inequality and the estimate for di, we have
|fij|‖eij‖∞ ≤ C(C
|fij |
2|λi|
n−1−2k
Thus, the series
ij fijeij converges to f in sup-norm, so Sp{eij , j = 1, 2, ..., di ; i =
1, 2, ...} is dense in sup-norm in C∞(M), hence in C(M) as well. ✷
Let us denote Sp{eij , j = 1, ..., di; i ≥ 1} by A
0 from now on. We shall
now show that C∞(M) can be replaced by the smaller subspace A∞0 in
Theorem 2.2. We need a lemma for this, which will be useful later on too.
Lemma 2.4 Let H1,H2 be Hilbert spaces and for i = 1, 2, let Li be (possibly
unbounded) self-adjoint operator on Hi with compact resolvents, and let Vi
be the linear span of eigenvectors of Li. Moreover, assume that there is an
eigenvalue of Li for which the eigenspace is one-dimensional, say spanned by
a unit vector ξi. Let Ψ be a linear map from V1 to V2 such that L2Ψ = ΨL1
and Ψ(ξ1) = ξ2. Then we have
〈ξ2,Ψ(x)〉 = 〈ξ1, x〉 ∀x ∈ V1. (1)
Proof:
By hypothesis on Ψ, it is clear that there is a common eigenvalue, say λ0, of
L1 and L2, with the eigenvectors ξ1 and ξ2 respectively. Let us write the set
of eigenvalues of Li as a disjoint union {λ0}
Λi (i = 1, 2), and let the corre-
sponding orthogonal decomposition of Vi be given by Vi = Cξi
Vλi ≡
Cξi ⊕ V
i, say, where V
i denotes the eigenspace of Li corresponding to the
eigenvalue λ. By assumption, Ψ maps Vλ1 to V
2 whenever λ is an eigenvalue
of L2, i.e. V
2 6= {0}, and otherwise it maps V
1 into {0}. Thus, Ψ(V
1) ⊆ V
Now, (1) is obviously satisfied for x = ξ1, so it is enough to prove (1) for all
x ∈ V ′1. But we have 〈ξ, x〉 = 0 for x ∈ V
1, and since Ψ(x) ∈ V
2 = V2
it follows that 〈ξ2,Ψ(x)〉 = 0 = 〈ξ1, x〉. ✷
Lemma 2.5 Let Y and α be as in Theorem 2.2. Then the following are
equivalent.
(a) For every y ∈ Y , ξy is smooth isometric.
(b) For every state φ on C(Y ), we have αφ(A
0 ) ⊆ A
0 , and αφL = Lαφ on
A∞0 .
Proof:
We prove only the nontrivial implication (b) ⇒ (a). Assume (b) that αφ
leaves A∞0 invariant and commutes with L on it, for every state φ. To
prove that α is a smooth isometric action, it is enough (see the proof of
Theorem 2.2) to prove that αy(A
∞) ⊆ A∞ for all y ∈ Y , where αy(f) :=
(id⊗evy)(f) = f ◦ξy, evy being the evaluation at the point y. LetM1, ...,Mk
be the connected components of the compact manifoldM . Thus, the Hilbert
space L2(M,dvol) admits an orthogonal decomposition ⊕ki=1L
2(Mi,dvol),
and the Laplacian L is of the form ⊕iLi where Li denotes the Laplacian on
Mi. Since each Mi is connected, we have Ker(Li) = Cχi, where χi is the
constant function on Mi equal to 1. Now, we note that for fixed y and i,
the image of Mi under the continuous function ξy must be mapped into a
component, sayMj. Thus, by applying Lemma 2.4 with H1 = L
2(Mi),H2 =
L2(Mj), Ψ = ξy and the L
2-continuity of the map f 7→ αy(f) = f ◦ ξy, we
αy(f)(x)dvol(x) =
f(x)dvol(x)
for all f in the linear span of eigenvectors of Li, hence (by density) for all f
in L2(Mi). It follows that
αy(f)dvol =
fdvol for all f ∈ L2(M), in
particular for all f ∈ C(M). Since αy is a ∗-homomorphism on C(M), we
〈αy(f), αy(g)〉 =
αy(fg)dvol =
fgdvol = 〈f, g〉,
for all f, g ∈ C(M). Thus, αy extends to an isometry on L
2(M), to be
denoted by the same notation, which by our assumption commutes with the
self-adjoint operator L on the core A∞0 , and hence αy commutes with L
n for
all n. In particular it leaves invariant the domains of each Ln, which implies
∞) ⊆ A∞. ✷
In view of the fact that the set of isometries of M , denoted by ISO(M),
is a compact second countable (i.e. compact metrizable) group, we see that
ISO(M) is the maximal compact second countable group acting on M such
that the action is smooth and isometric. In other words, if we consider a
catogory whose objects are compact metrizable groups acting smoothly and
isometrically on M , and morphisms are the group homomorphisms com-
muting with the actions on M , then ISO(M) (with its canonical action on
M) is the initial object of this cateogory. However, one can take a more
general viewpoint and consider the category of compact metrizable spaces
Y equipped with a continuous map θ : M × Y → M satisfying (i)-(iii) of
Theorem 2.2, or equivalently, the pair of commutative unital C∗-algebras
B = C(Y ) and a unital C∗-homomorphism α : C(M) → C(M) → B satisfy-
ing the conditions (i)-(iii). The set of isometries ISO(M) (as a topological
space) can be identified with the universal object of this category, and then
one can prove that it has a group structure.
It is quite natural to formulate a quantum analogue of the above, by con-
sidering, in the spirit of Woronowicz and Soltan (see [19] and [13]), ‘quantum
families of isometries’, which can be defined to be a pair (B, α) where B is
a (not necessarily commutative) C∗-algebra and α : C(M) → C(M)⊗ B is
unital C∗-homomorhism satisfying (i)-(iii) of Theorem 2.2, i.e. the linear
span of α(C(M))(1⊗B) (which is not necessarily a ∗-subalgebra any more,
B being possibly noncommutative) is norm-dense in C(M)⊗ B and for ev-
ery state φ on B, the map αφ keeps C
∞(M) invariant and commutes with
the Laplacian L. The morphisms of this category are obvious. We shall
prove that this category has a universal object, and this universal object
can be equipped with a canonical quantum group structure. This will define
the quantum isometry group of a manifold. However, we shall go beyond
classical manifolds and define quantum isometry group QISO(A∞,H,D)
for a spectral triple (A∞,H,D), with A∞ being unital, and satisfying cer-
tain assumptions. To this end, we need to carefully formulate the notion
of Laplacian in noncommutative geometry, which is the goal of the next
subsection.
2.2 Laplacian in noncommutative geometry
Given a spectral triple (A∞,H,D), we recall from [10] and [6] the con-
struction of the space of one-forms. We have a derivation from A∞ to the
A∞-A∞ bimodule B(H) given by a 7→ [D, a]. This induces a bimodule
morphism π from Ω1(A∞) (the bimodule of universal one-forms on A∞) to
B(H), such that π(δ(a)) = [D, a], where δ : A∞ → Ω1(A∞) denotes the
universal derivation map. We set Ω1D ≡ Ω
∞) := Ω1(A∞)/Ker(π) ∼=
π(Ω1(A∞)) ⊆ B(H). Assume that the spectral triple is of compact type
and has a finite dimension in the sense of Connes ([6]), i.e. there is some
p > 0 such that the operator |D|−p (interpreted as the inverse of the re-
striction of |D|p on the closure of its range, which has a finite co-dimension
since D has compact resolvents) has finite nonzero Dixmier trace, denoted
by Trω (where ω is some suitable Banach limit, see, e.g. [6], [10]). Con-
sider the canonical ‘volume form’ τ coming from the Dixmier trace, i.e.
τ : B(H) → C defined by τ(A) := 1
Trω(|D|−p)
Trω(A|D|
−p). Let us at this
point assume that the spectral triple is QC∞, i.e. A∞ and {[D, a], a ∈ A∞}
are contained in the domains of all powers of the derivation [|D|, ·]. Under
this assumption, τ is a positive faithful trace on the C∗-subalgebra gener-
ated by A∞ and {[D, a] a ∈ A∞}, and the GNS Hilbert space L2(A∞, τ) is
denoted by H0D. Similarly, we equip Ω
D with a semi-inner product given by
< η, η′ >:= τ(η∗η′), and denote the Hilbert space obtained from it by H1D.
The map dD : H
D → H
D given by dD(·) = [D, ·] is an unbounded densely
defined linear map. Let us assume the following:
Assumption(i) (a) dD is closable (the closure is denoted again by dD);
(b) A∞ ⊆ Dom(L), where L := −d∗DdD and A
∞ is viewed as a dense sub-
space of H0D;
At this point, let us show that this assumption is valid under a very
natural condition on the spectral triple.
Lemma 2.6 Suppose that for every element a ∈ A∞, the map R ∋ t 7→
αt(X) := exp(itD)Xexp(−itD) is differentiable at t = 0 in the norm-
topology of B(H), where X = a or [D, a]. Then the assumption (i) is sat-
isfied. Moreover, in this case, L maps A∞ into the weak closure of A∞ in
B(H0D).
Proof :
We first observe that τ(αt(A)) = τ(A) for all t and for all A ∈ B(H), since
exp(itD) commutes with |D|−p. If moreover, A belongs to the domain of
norm-differentiability (at t = 0) of αt, i.e.
αt(A)−A
→ i[D,A] in operator-
norm, then it follows from the property of the Dixmier trace that τ([D,A]) =
limt→0
τ(αt(A))−τ(A)
= 0. Now, since by assumption we have the norm-
differentiability at t = 0 of αt(A) for A belonging to the ∗-subalgebra (say
B) generated by A∞ and [D,A∞], it follows that τ([D,A]) = 0 ∀A ∈ B. Let
us now fix a, b, c ∈ A∞ and observe that
< a dD(b), dD(c) >
= τ((a dD(b))
∗dD(c) >
= −τ([D, [D, b∗]a∗c]) + τ([D, [D, b∗]a∗]c)
= τ([D, [D, b∗]a∗]c),
using the fact that τ([D, [D, b∗]a∗c]) = 0. This implies
| < a dD(b), dD(c) > | ≤ ‖[D, [D, b
∗]a∗]‖τ(c∗c)
2 = ‖[D, [D, b∗]a∗]‖‖c‖2,
where ‖c‖2 = τ(c
2 denotes the L2-norm of c ∈ H0D. This proves that
a dD(b) belongs to the domain of d
D for all a, b ∈ A
∞, so in particular
d∗D is dense, i.e. dD is closable. Moreover, taking a = 1, we see that
∞) ⊆ Dom(d∗D), or in other words, A
∞ ⊆ Dom(d∗DdD). This proves
(i)(a) and (i)(b). The last sentence in the statement of the lemma can be
proven along the line of Theorem 2.9, page 129, [10]. ✷
We need few more assumptions on the operator L to define the quantum
isometry group.
Assumption (ii): L has compact resolvents,
Assumption(iii): L(A∞) ⊆ A∞;
Assumption(iv): Each eigenvector of L (which has a discrete spectrum,
hence a complete set of eigenvectors) belongs to A∞;
Assumption(v)(‘connectedness assumption’): the kernel of L is one-dimensional,
spanned by the identity 1 of A∞, viewed as a unit vector in H0D.
We call L the noncommutative Laplacian and Tt the noncommutative heat
semigroup. We summarize some simple observations in form of the following
Lemma 2.7 (a) If the assumptions (i)-(v) are valid, then for x ∈ A∞, we
have L(x∗) = (L(x))∗.
(b) If Tt := exp(tL) maps H
D into A
∞ for all t > 0, the the assumption
(iv) is satisfied.
Proof :
It follows by simple calculation using the facts that τ is a trace and dD(x
−(dD(x))
∗ that
τ(L(x∗)∗y)
= −τ(dD(x)dD(y)) = −τ(dD(y)dD(x)) = τ((dD(y
∗))∗dD(x))
= < y∗,L(x) >= τ(yL(x)) = τ(L(x)y),
for all y ∈ A∞. By density of A∞ in H0D (a) follows. To prove (b), we note
that if x ∈ H0D is an eigenvector of L, say L(x) = λx (λ ∈ C), then we have
Tt(x) = e
λtx, hence x = e−λtTt(x) ∈ A
Since by assumption, L has a countable set of eigenvalues each with finite
multiplicity, let us denote them by λ0 = 0, λ1, λ2, ... with V0 = C 1, V1, V2, ...
be corresponding eigenspaces (finite dimensional), and for each i, let {eij , j =
1, ..., di} be an orthonormal basis of Vi. By Assumption (iv), Vi ⊆ A
∞ for
each i, Vi is closed under ∗, and moreover, {e
ij , j = 1, ..., di} is also an or-
thonormal basis for Vi, since τ(x
∗y) = τ(yx∗) for x, y ∈ A∞. We also make
the following
Assumption (vi) The complex linear span of {eij , i = 0, 1, ...; j = 1, ..., di},
say A∞0 , is norm-dense in A
Definition 2.8 We say that a spectral triple satisfying the assumptions (i)-
(vi) admissible.
Remark 2.9 We have just seen that classical spectral triple (A∞ = C∞(M),H,D),
where M is compact connected spin manifold, H is the L2 space of square
integrable spinors and D is the Dirac operator, is indeed admissible in our
sense. Later on we shall discuss how we can weaken the connectedness
assumption as well, thus accommodating a general classical (commutative)
spectral triple in our set-up. Moreover, the standard examples of noncom-
mutative spectral triples, e.g. those on Aθ, quantum Heisenberg manifold
etc., do belong to the admissible class.
Lemma 2.10 Let us assume that the spectral triple (A∞,H,D) is admis-
sible. Let Ψ : A∞0 → A
0 be a (norm-) bounded linear map, such that
Ψ(1) = 1, and Ψ ◦L = L◦Ψ on the subspace A∞0 spanned (algebraically) by
Vi, i = 1, 2, .... Then τ(Ψ(x)) = τ(x) for all x ∈ A
Proof :
By Lemma 2.4 with H1 = H2 = H
D, ξ1 = ξ2 = 1, we have τ(Ψ(x)) = τ(x)
for all x ∈ A∞0 . By the norm-continuity of Ψ and τ it extends to the whole
of A∞. ✷
2.3 Definition and existence of the quantum isometry group
We begin by recalling the definition of compact quantum groups and their
actions from [18]. A compact quantum group is given by a pair (S,∆), where
S is a unital separable C∗ algebra equipped with a unital C∗-homomorphism
∆ : S → S ⊗ S (where ⊗ denotes the injective tensor product) satisfying
(ai) (∆⊗ id) ◦∆ = (id⊗∆) ◦∆ (co-associativity), and
(aii) the linear span of ∆(S)(S ⊗ 1) and ∆(S)(1 ⊗ S) are norm-dense in
S ⊗ S.
It is well-known (see [18]) that there is a canonical dense ∗-subalgebra S0
of S, consisting of the matrix coefficients of the finite dimensional unitary
(co)-representations of S, and maps ǫ : S0 → C (co-unit) and κ : S0 → S0
(antipode) defined on S0 which make S0 a Hopf ∗-algebra.
We say that the compact quantum group (S,∆) acts on a unital C∗
algebra B, if there is a unital C∗-homomorphism α : B → B ⊗ S satisfying
the following :
(bi) (α⊗ id) ◦ α = (id⊗∆) ◦ α, and
(bii) the linear span of α(B)(1 ⊗ S) is norm-dense in B ⊗ S.
Let us now recall the concept of universal quantum groups as in [17],
[15] and references therein. We shall use most of the terminologies of [15],
e.g. Woronowicz C∗ -subalgebra, Woronowicz C∗-ideal etc, however with
the exception that we shall call the Woronowicz C∗ algebras just compact
quantum groups, and not use the term compact quantum groups for the
dual objects as done in [15]. For Q ∈ GLn(C), let Au(Q) denote the uni-
versal compact quantum group generated by uij, i, j = 1, ..., n satisfying the
relations
uu∗ = In = u
∗u, u′QuQ−1 = In = QuQ
−1u′,
where u = ((uij)), u
′ = ((uji)) and u = ((u
ij)). The coproduct, say ∆̃, is
given by,
∆̃(uij) =
uik ⊗ ukj.
We refer the reader to [17] for a detailed discussion on the structure and clas-
sification of such quantum groups. Let us denote by Ui the quantum group
Adi(I), where di is dimension of the subspace Vi. We fix a representation
βi : Vi → Vi⊗Ui of Ui on the Hilbert space Vi, given by βi(eij) =
k eik⊗u
for j = 1, ..., di, where u
(i) ≡ u
are the generators of Ui as discussed before.
Thus, both u(i) and ¯u(i) are unitaries. It follows from [15] that the represen-
tations βi canonically induce a representation β = ∗iβi of the free product
U := ∗iUi (which is a compact quantum group, see [15] for the details) on
the Hilbert space H0D, such that the restriction of β on Vi coincides with βi
for all i.
In view of the characterization of smooth isometric action on a classical
manifold, we make the following definitions.
Definition 2.11 A quantum family of smooth isometries of a noncommu-
tative manifold A∞ (or, more precisely on the corresponding spectral triple)
is a pair (S, α) where S is a separable unital C∗-algebra, α : A → A ⊗ S
(where A denotes the C∗ algebra obtained by completing A∞ in the norm of
B(H0D)) is a unital C
∗-homomorphism, satisfying the following:
(a) Sp
α(A)(1⊗ S)
= A⊗ S,
(b) αφ := (id ⊗ φ) ◦ α maps A
0 into itself and commutes with L on A
for every state φ on S.
In case the C∗-algebra S has a coproduct ∆ such that (S,∆) is a compact
quantum group and α is an action of (S,∆) on A, we say that (S,∆) acts
smoothly and isometrically on the noncommutative manifold.
Fix a spectral triple (A∞,H,D). Consider the category Q with the
object-class consisting of all quantum families of isometries (S, α) of the
given noncommutative manifold, and the set of morphismsMor((S, α), (S ′, α′))
being the set of unital C∗-homomorphisms φ : S → S ′ satisfying (id⊗φ)◦α =
α′. We also consider another category Q′ whose objects are triplets (S,∆, α)
where (S,∆) is a compact quantum group acting smoothly and isometrically
on the given noncommutative manifold, with α being the corresponding ac-
tion. The morphisms are the homomorphisms of compact quantum groups
which are also morphisms of the underlying quantum families. The forget-
ful functor F : Q′ → Q is clearly faithful, and we can view F (Q′) as a
subcategory of Q.
Let us assume from now on that the spectral triple (A∞,H,D) is admis-
sible. Our aim is to prove the existence of a universal object in Q. We shall
also prove that the (unique upto isomorphism) universal object belongs to
F (Q′), and its pre-image in Q′ is a universal object in the category Q′. To
this end, we need some preparatory results.
Lemma 2.12 Consider an admissible spectral triple (A∞,H,D) and let
(S, α) be a quantum family of smooth isometries of the spectral triple. More-
over, assume that the action α is faithful in the sense that there is no proper
C∗-subalgebra S1 of S such that α(A
∞) ⊆ A∞ ⊗ S1. Then α̃ : A
∞ ⊗ S →
A∞ ⊗ S defined by α̃(a⊗ b) : α(a)(1 ⊗ b) extends to an S-linear unitary on
the Hilbert S-module H0D ⊗S, denoted again by α̃. Moreover, we can find a
C∗-isomorphism φ : U/I → S between S and a quotient of U by a C∗-ideal
I of U , such that α = (id ⊗ φ) ◦ (id ⊗ ΠI) ◦ β on A
∞ ⊆ H0D, where ΠI
denotes the quotient map from U to U/I.
If, furthermore, there is a compact quantum group structure on S given
by a coproduct ∆ such that (S,∆, α) is an object in Q′, the map α : A∞ →
A∞⊗S extends to a unitary representation (denoted again by α) of the com-
pact quantum group (S,∆) on H0D. In this case, the ideal I is a Woronowicz
C∗-ideal and the C∗-isomorphism φ : U/I → S is a morphism of compact
quantum groups.
Proof :
Let ω be any state on S. Since the action α : A∞ → A∞ ⊗ S is smooth
and isometric, we conclude by Lemma 2.10 that τ(αω(x)) = τ(x)ω(1) for all
x ∈ A. Since ω is arbitrary, we have (τ ⊗ id)α(x) = τ(x)1S for all x ∈ A.
So, < α(x), α(y) >S=< x, y > 1S , where < ·, · >S denotes the S-valued
inner product of the Hilbert module H0D ⊗S. This proves that α̃ defined by
α̃(x⊗ b) := α(x)(1⊗ b) (x ∈ A∞, b ∈ S) extends to an S-linear isometry on
the Hilbert S-module H0D⊗S. Moreover, since α(A
∞)(1⊗S) is norm-dense
in Ā⊗S, it is clear that the S-linear span of the range of α(A∞) is dense in
the Hilbert module H0D ⊗ S, or in other words, the isometry α̃ has a dense
range, so it is a unitary.
Since αω leaves each Vi invariant, it is clear that α maps Vi into Vi ⊗ S
for each i. Let v
(j, k = 1, ..., di) be the elements of S such that α(eij) =
k eik ⊗ v
. Note that vi := ((v
)) is a unitary in Mdi(C)⊗S. Moreover,
the ∗-subalgebra generated by all {v
, i, j, k ≥ 1} must be dense in S by
the assumption of faithfulness.
We have already remarked that {e∗ij} is also an orthonormal basis of
Vi, and since α, being a C
∗-action on A, is ∗-preserving, we have α(e∗ij) =
(α(eij))
, and therefore ((v
)) is also unitary. By univer-
sality of Ui, there is a C
∗-homomorphism from Ui to S sending u
and by definition of the free product, this induces a C∗-homomorphism, say
Π, from U onto S, so that U/I ∼= S, where I := Ker(Π).
In case S has a coproduct ∆ making it into a compact quantum group
and α is a quantum group action, it is easy to see that the subalgebra of
S generated by v
is a Hopf algebra, with ∆(v
. From
this, it follows that Π is Hopf-algebra morphism, hence I is a Woronowicz
C∗-ideal. ✷
Before we state and prove the main theorem, let us note the following
elementary fact about C∗-algebras.
Lemma 2.13 Let C be a C∗ algebra and F be a nonempty collection of
C∗-ideals (closed two-sided ideals) of C. Then for any x ∈ C, we have
‖x+ I‖ = ‖x+ I0‖,
where I0 denotes the intersection of all I in F and ‖x + I‖ = inf{‖x −
y‖ : y ∈ I} denotes the norm in C/I.
Proof :
It is clear that supI∈F ‖x + I‖ defines a norm on C/I0, which is in fact a
C∗-norm since each of the quotient norms ‖ · +I‖ is so. Thus the lemma
follows from the uniqueness of C∗ norm on the C∗ algebra C/I0. ✷
Theorem 2.14 For any admissible spectral triple (A∞,H,D), the category
Q of quantum families of smooth isometries has a universal (initial) object,
say (G, α0). Moreover, G has a coproduct ∆0 such that (G,∆0) is a com-
pact quantum group and (G,∆0, α0) is a universal object in the category Q
of compact quantum groups acting smoothly and isometrically on the given
spectral triple. The action α0 is faithful.
Proof :
Recall the C∗-algebra U considered before, and the map β from H0D to
H0D⊗U . By our definition of β, it is clear that β(A
0 ) ⊆ A
0 ⊗algU . However,
β is only a linear map (unitary) but not necessarily a ∗-homomorphism. We
shall construct the universal object as a suitable quotient of U . Let F
be the collection of all those C∗-ideals I of U such that the composition
ΓI := (id⊗ΠI) ◦ β : A
0 → A
0 ⊗alg (U/I) extends to a C
∗-homomorphsim
from Ā to Ā ⊗ (U/I), where ΠI denotes the quotient map from U onto
U/I. This collection is nonempty, since the trivial one-dimensional C∗-
algebra C gives an object in Q and by Lemma 2.12 we do get a member of
F . Now, let I0 be the intersection of all ideals in F . We claim that I0 is
again a member of F . Since any C∗-homomorphism is contractive, we have
‖ΓI(a)‖ ≡ ‖β(a) + Ā ⊗ I‖ ≤ ‖a‖ for all a ∈ A
0 and I ∈ F . By Lemma
2.13, we see that ‖ΓI0(a)‖ ≤ ‖a‖ for a ∈ A
0 , so ΓI0 extends to a norm-
contractive map on Ā by the density of A∞0 in Ā. Moreover, for a, b ∈ Ā
and for I ∈ F , we have ΓI(ab) = ΓI(a)ΓI(b). Since ΠI = ΠI ◦ ΠI0 , we can
rewrite the homomorphic property of ΓI as
ΓI0(ab)− ΓI0(a)ΓI0(b) ∈ Ā ⊗ (I/I0).
Since this holds for every I ∈ F , we conclude that ΓI0(ab)−ΓI0(a)ΓI0(b) ∈
I∈F Ā⊗(I/I0) = (0), i.e. ΓI0 is a homomorphism. In a similar way, we can
show that it is a ∗-homomorphism. Since each βi is a unitary representation
of the compact quantum group Ui on the finite dimensional space Vi, it
follows that βi(Vi)(1 ⊗ Ui) is total in Vi ⊗ Ui. In particular, for any vi ∈ Vi
(i arbitrary), the element vi ⊗ 1Ui = vi ⊗ 1U belongs to the linear span of
βi(Vi)(1⊗Ui) ⊂ β(Vi)(1⊗U). Thus, A
0 ⊗1U is contained in the linear span of
β(A∞0 )(1⊗U) and henceA
0 ⊗1 U
is linearly spanned by ΓI0(A
0 )(1⊗U/I0).
By the norm-denisty of A∞0 in A and the contractivity of the quotient map,
it follows that A ⊗ U/I0 is the closed linear span of ΓI0(A
0 )(1 ⊗ U/I0).
This completes the proof that (U/I0,ΓI0) is indeed an object of Q.
We now show that G := U/I0 is a universal object in Q. To see this, con-
sider any object (S, α) of Q. Without loss of generality we can assume the
action to be faithful, since otherwise we can replace S by the C∗-subalgebra
generated by the elements {v
} appearing in the proof of Lemma 2.12. But
by Lemma 2.12 we can further assume that S is isomorphic with U/I for
some I ∈ F . Since I0 ⊆ I, we have a C
∗-homomorphism from U/I0 onto
U/I, sending x+I0 to x+I, which is clearly a morphism in the category Q.
This is indeed the unique such morphism, since it is uniquely determined on
the dense subalgebra generated by {u
+ I0, i, j, k ≥ 1} of G.
To construct the coproduct on G = U/I0, we first consider α
(2) = (ΓI0 ⊗
id) ◦ΓI0 : A → A⊗G ⊗G. It is easy to verify that (G ⊗G, α
(2)) is an object
in the category Q, so by the universality of (G,ΓI0), we have a unique unital
C∗-homomorphism ∆0 : G → G ⊗ G satisfying
(id⊗∆0) ◦ ΓI0(x) = α
(2)(x) ∀x ∈ A.
Taking x = eij, we get
eil ⊗ (πI0 ⊗ πI0)
eil ⊗∆0(πI0(u
Comparing coefficients of eil, and recalling that ∆̃(u
(where ∆̃ denotes the coproduct on U), we have
(πI0 ⊗ πI0) ◦ ∆̃ = ∆0 ◦ πI0 (2)
on the linear span of {u
, i, j, k ≥ 1}, and hence on the whole of U . This
implies that ∆0 maps I0 = Ker(πI0) into Ker(πI0⊗πI0) = (I0⊗1+1⊗I0) ⊂
U ⊗ U . In other words, I0 is a Hopf C
∗-ideal, and hence G = U/I0 has the
canonical compact quantum group structure as a quantum subgroup of U . It
is clear from the relation (2) that ∆0 coincides with the canonical coproduct
of the quantum subgroup U/I0 inherited from that of U . It is also easy to
see that the object (G,∆0,ΓI0) is universal in the category Q
′, using the fact
that (by Lemma 2.12) any compact quantum group (G,Φ) acting smoothly
and isometrically on the given spectral triple is isomorphic with a quantum
subgroup U/I, for some Hopf C∗-ideal I of U .
Finally, the faithfulness of α0 follows from the universality by standard
arguments which we briefly sketch. If G1 ⊂ G is a ∗-subalgebra of G such
that α0(A) ⊆ A ⊗ G1, it is easy to see that (G1,∆0, α0) is also a universal
object, and by definition of universality of G it follows that there is a unique
morphism, say j, from G to G1. But the map j ◦ i is a morphism from G to
itself, where i : G1 → G is the inclusion. Again by universality, we have that
j ◦ i = idG , so in particular, i is onto, i.e. G1 = G. ✷
Definition 2.15 We shall call the universal object (G,∆0) obtained in the
theorem above the quantum isometry group of (A∞,H,D) and denote it
by QISO(A∞,H,D), or just QISO(A∞) (or sometimes QISO(Ā)) if the
spectral triple is understood from the context.
Remark 2.16 Assume that an admissible spectral triple (A∞,H,D) also
satisfies the condition (i) of Lemma 2.5, i.e.
Dom(Ln) = A∞. Let α :
A → A⊗S be a smooth isometric action on A∞ by a compact quantum group
S. We recall from the proof of Lemma 2.12 that the map α̃ from A⊗ S to
itself extends to an S-linear unitary on the Hilbert S-module H0D ⊗ S, i.e.
α̃ can be viewed as a unitary in B(H0D) ⊗ S. Clearly, for any state φ on
S, we have αφ = (id ⊗ φ)(α̃) ∈ B(H
D). Now, by the definition of a smooth
isometric action, the bounded operator αφ commutes with the self-adjoint
operator L on A∞0 , which is a core for L. So, αφ must commute with L
for all n, and in particular keeps A∞ =
nDom(L
n) invariant.
Remark 2.17 Let us now briefly indicate how one can weaken the hy-
pothesis of connectedness. Such an extension of our results is desirable
to accommodate the classical spaces, including the finite sets and graphs,
in our framework. One possibile approach could be to consider the cate-
gory of compact quantum group actions α which are not only ‘smmoth’ and
‘isometric’ in our sense, but also satisfy the τ -invariance condition, i.e.
(τ ⊗ id)(α(a)) = τ(a)1. It is easy to see that the connectedness assumption
has been used by us only to prove that the τ -invariance is automatic for
smooth isometric actions. Thus, if we work in the smaller cateogory of such
τ -invraiant actions only, the proof of Theorem 2.14 does go through and thus
we can prove the existence of a universal object, to be defined as the quantum
isometry group. It is easy to see that for the algebra of functions on a finite
set, with the spectral triple given by D = 0, this quantum isometry group
coincides with thw quantum permutation group defined by Wang.
Remark 2.18 It is easy to see how to extend our formulation and results
to spectral triples which are not necessarily of type II, i.e. when the trace
τ is replaced by some non-tracial positive functional. Indeed, our construc-
tion will go through in such a situation more or less verbatim, by replacing
the universal quantum groups Adi(I) by Adi(Qi) for some suitable choice of
matrices Qi coming from the modularity property of τ .
2.4 Construction of quantum group-equivariant spectral triples
In this subsection, we shall briefly discuss the relevance of quantum isometry
group to the problem of constructing quantum group equivariant spectral
triples, which is important to understand the role of quantum groups in the
framework of noncommutative geometry. There has been a lot of activity
in this direction recently, see, for example, the articles by Chakraborty and
Pal ([5]), Connes ([7]), Landi et al ([8]) and the references therein. In the
classical situation, there exists a natural unitary representation of the isom-
etry group G = ISO(M) of a manifold M on the Hilbert space of forms, so
that the operator d+d∗ (where d is the de-Rham differential operator) com-
mutes with the representation. Indeed, d+d∗ is also a Dirac operator for the
spectral triple given by the natural representation of C∞(M) on the Hilbert
space of forms, so we have a canonical construction of G-equivariant spectral
triple. Our aim in this subsection is to generalize this to the noncommuta-
tive framework, by proving that dD + d
D is equivariant with respect to a
canonical unitary representation on the Hilbert space of ‘noncommutative
forms’ (see, for example, [10] for a detailed discussion of such forms).
Consider an admissible spectral triple (A∞,H,D) and moreover, make
the assumption of Lemma 2.6, i.e. assume that t 7→ eitDxe−itD is norm-
differentiable at t = 0 for all x in the ∗-algebra B generated by A∞ and
[D,A∞].
Lemma 2.19 In the notation of Lemma 2.6, we have the following (where
b, c ∈ A∞):
d∗D(dD(b)c) = −
(bL(c)− L(b)c− L(bc)) . (3)
Proof:
Denote by χ(b, c) the right hand side of euqation (3) and fix any a ∈ A∞.
Using the facts the the functional τ is a faithful trace on the ∗-algebra B,
L = −d∗DdD and that [D,X] = 0 for any X in B, we have,
τ(a∗χ(b, c))
{τ(a∗bL(c)) + τ(ca∗L(b)) + τ(a∗L(bc))}
{τ([D, a∗b][D, c]) − τ([D, ca∗][D, b])− τ([D, a∗][D, bc])}
{τ(a∗[D, b][D, c]) − τ([D, c]a∗[D, b])− τ(c[D, a∗][D, b])− τ([D, a∗][D, b]c)}
= −τ([D, a∗][D, b]c)
= τ([D, a]∗[D, b]c)
= 〈dD(a), dD(b)c〉
= τ(a∗(d∗D(dD(b)c))).
From this, we get the following by a simple computation:
〈adD(b), a
′dD(b
′)〉 = −
τ(b∗Ψ(a∗a, b′)), (4)
for a, b, a′, b′ ∈ A∞, and where Ψ(x, y) := L(xy)−L(x)y+xL(y). Now, let us
denote the quantum isometry group of the given spectral triple (A∞,H,D)
by (G,∆, α). Let A0 denote the ∗-algebra generated by A
0 , G0 denote ∗-
algebra of G generated by matrix elements of irreducible representations.
Clearly, α : A0 → A0 ⊗alg G0 is a Hopf-algebraic action of G0 on A0. Define
Ψ̃ : (A0 ⊗alg G0)× (A0 ⊗alg G0) → A0 ⊗alg G0 by
Ψ̃((x⊗ q), (x′ ⊗ q′)) := Ψ(x, x′)⊗ (qq′).
It follows from the relation (L ⊗ id) ◦ α = α ◦ L on A0 that
Ψ̃(α(x), α(y)) = α(Ψ(x, y)). (5)
We now define a linear map α(1) from the linear span of {adD(b) : a, b ∈ A0}
to H1D ⊗ G by setting
α(1)(adD(b)) :=
i dD(b
j )⊗ a
where for any x ∈ A0 we write α(x) =
i ∈ A0⊗algG0 (summation
over finitely many terms). We shall sometimes use the Sweedler convention
of writing the above simply as α(x) = x(1) ⊗ x(2). It then follows from the
identities (4) and (5), and also the fact that (τ ⊗ id)(α(a)) = τ(a)1 for all
a ∈ A0 that
〈adD(b), a
′dD(b
(τ ⊗ id)(α(b∗)Ψ̃(α(a∗a′), α(b′)))
(τ ⊗ id)(α(b∗)α(Ψ(a∗a′, b′)))
(τ ⊗ id)(α(b∗Ψ(a∗a′, b′)))
τ(b∗Ψ(a∗a′, b′))1G
= 〈adD(b), a
′dD(b
′)〉1G .
This proves that α(1) is indeed well-defined and extends to a G-linear isom-
etry on H1D ⊗ G, to be denoted by U
(1), which sends (adD(b)) ⊗ q to
α(1)(adD(b))(1 ⊗ q), a, b ∈ A0, q ∈ G. Moreover, since the linear span
of α(A∞0 )(1 ⊗ G) is dense in H
D ⊗ G, it is easily seen that the range of the
isometry U (1) is the whole of H1D ⊗G, i.e. U
(1) is a unitary. In fact, from its
definition it can also be shwon that U (1) is a unitary representation of the
compact quantum group G on H1D.
In a similar way, we can construct unitary representation U (n) of G on
the Hilbert space of n-forms for any n ≥ 1, by defining
U (n)((a0dD(a1)dD(a2)...dD(an))⊗q) = a
0 dD(a
1 )...dD(a
n )⊗(a
1 ...a
n q), ai ∈ A
(using Sweedler convention) and verifying that it extends to a unitary.
We also denote by U (0) the unitary representation α̃ on H0D discussed be-
fore. Finally, we have a unitary representation U =
n≥0 U
(n) of G on
H̃ :=
D, and also extend dD as a closed densely defined operator on H̃
in the obvious way, by defining dD(a0dD(a1)...dD(an)) = dD(a0)...dD(an).
It is now straightforward to see the following:
Theorem 2.20 The operator D′ := dD+d
D is equivariant in the sense that
U(D′ ⊗ 1) = (D′ ⊗ 1)U .
We point out that there is a natural representation π of A on H̃ given
by π(a)(a0dD(a1)...dD(an)) = aa0dD(a1)...dD(an), and (π(A
∞), H̃,D′) is
indeed a spectral triple, which is G-equivariant.
Although the relation between spectral properties of D and D′ is not
clear in general, in many cases of interest (e.g. when there is an underlying
type (1, 1) spectral data in the sense of [10]) these two Dirac operators are
closely related. As an illustration, consider the canonical spectral on the
noncommutative 2-torus Aθ, which is discussed in some details in the next
section. In this case, the Dirac operator D acts on L2(Aθ, τ) ⊗ C
2, and it
can easily be shown (see [10]) that the Hilbert space of forms is isomorphic
with L2(Aθ, τ)⊗C
4 ∼= L2(Aθ)⊗C
2; thus D′ is essentially same as D in this
case.
3 Examples and computations
We give some simple yet interesting explicit examples of quantum isometry
groups here. However, we give only some computational details for the first
example, and for the rest, the reader is referred to a companion article ([3]).
Example 1 : commutative tori
Consider M = T, the one-torus, with the usual Riemannian structure. The
∗-algebra A∞ = C∞(M) is generated by one unitary U , which is the multi-
plication operator by z in L2(T). The Laplacian is given by L(Un) = −n2Un.
If a compact quantum group (S,∆S) acts on A
∞ smoothly, let An, n ∈ Z be
elements of S such that α0(U) =
n ⊗An (here α0 : A
∞ → A∞ ⊗alg S
is the S-action on A∞). Note that this infinite sum converges at least
in the topology of the Hilbert space L2(T) ⊗ L2(S), where L2(S) denotes
the GNS space for the Haar state of S. It is clear that the condition
(L ⊗ id) ◦ α0 = α0 ◦ L forces to have An = 0 for all but n = ±1. The
conditions α0(U)α0(U)
∗ = α0(U)
∗α0(U) = 1 ⊗ 1 further imply the follow-
A∗1A1 +A
−1A−1 = 1 = A1A
1 +A−1A
A∗1A−1 = A
−1A1 = A1A
−1 = A−1A
1 = 0.
It follows that A±1 are partial isometries with orthogonal domains and
ranges. Say, A1 has domain P and range Q. Hence the domain and
range of A−1 are respectively 1 − P and 1 − Q. Consider the unitary
V = A + B, so that V P = A, V (1 − P ) = B. Now, from the fact that
(L⊗ id)(α0(U
2)) = α0(L(U
2)) it is easy to see that the coefficient of 1⊗1 in
the expression of α0(U)
2 must be 0, i.e. AB+BA = 0. From this, it follows
that V and P commute and therefore P = Q. By straightforward calculation
using the facts that V is unitary, P is a projection and V and P commute, we
can verify that α0 given by α0(U) = U ⊗V P +U
−1⊗V (1−P ) extends to a
∗-homomorphsim from A∞ to A∞⊗C∗(V, P ) satisfying (L⊗id)◦α0 = α0◦L.
It follows that the C∗ algebra QISO(T) is commutative and generated by a
unitary V and a projection P , or equivalently by two partial isometries A,
B such that A∗A = AA∗, B∗B = BB∗, AB = BA = 0. So, as a C∗ algebra
it is isomorphic with C(T) ⊕ C(T) ∼= C(T × Z2). The coproduct (say ∆0)
can easily be calculated from the requirement of co-associativity, and the
Hopf algebra structure of QISO(T) can be seen to coincide with that of
the semi-direct product of T by Z2, where the generator of Z2 acts on T by
sending z 7→ z̄.
We summarize this in form of the following.
Theorem 3.1 The universal quantum group of isometries QISO(T) of the
one-torus T is isomorphic (as a quantum group) with C(T >⊳Z2) = C(ISO(T)).
We can easily extend this result to higher dimensional commutative tori,
and can prove that the quantum isometry group coincides with the classical
isometry group. This is some kind of rigidity result, and it will be interest-
ing to investigate the nature of quantum isometry groups of more general
classical manifolds.
Example 2 : Noncommutative torus; holomorphic isomrtries
Next we consider the simplest and well-known example of noncommutative
manifold, namely the noncommutative two-torus Aθ, where θ is a fixed
irrational number (see [6]). It is the universal C∗ algebra generated by
two unitaries U and V satisfying the commutation relation UV = λV U ,
where λ = e2πiθ. There is a canonical faithful trace τ on Aθ given by
τ(UmV n) = δmn. We consider the canonical spectral triple (A
∞,H,D),
where A∞ is the unital ∗-algebra spanned by U, V , H = L2(τ)⊕ L2(τ) and
D is given by
0 d1 + id2
d1 − id2 0
where d1 and d2 are closed unbounded linear maps on L
2(τ) given by
mV n) = mUmV n, d2(U
mV n) = nUmV n. It is easy to compute the
space of one-forms Ω1D (see [4], [10], [6]) and the Laplacian L = −d
∗d is
given by L(UmV n) = −(m2 + n2)UmV n. For simplicity of computation,
instead of the full quantum isometry group we at first concentrate on an
interesting quantum subgroup G = QISOhol(A∞,H,D), which is the uni-
versal quantum group which leaves invariant the subalgebra ofA∞ consisting
of polynomials in U , V and 1, i.e. span of UmV n with m,n ≥ 0. The proof
of existence and uniqueness of such a universal quantum group is more or
less identical to the proof of existence and uniqueness of QISO. We call G
the quantum group of “holomorphic” isometries, and observe in the theorem
stated below without proof (see [3]) that this quantum group is nothing but
the quantum double torus studied in [11].
Theorem 3.2 Consider the following co-product ∆B on the C
∗ algebra B =
C(T2)⊕A2θ, given on the generators A0, B0, C0,D0 as follows ( where A0,D0
correspond to C(T2) and B0, C0 correspond to A2θ)
∆B(A0) = A0 ⊗A0 + C0 ⊗B0, ∆B(B0) = B0 ⊗A0 +D0 ⊗B0,
∆B(C0) = A0 ⊗ C0 +C0 ⊗D0, ∆B(D0) = B0 ⊗ C0 +D0 ⊗D0.
Then (B,∆0) is a compact quantum group and it has an action α0 on Aθ
given by
α0(U) = U ⊗A0 + V ⊗B0, α0(V ) = U ⊗C0 + V ⊗D0.
Moreover, (B,∆B) is isomorphic (as quantum group) with G = QISO
hol(A∞,H,D).
We refer to [3] for a proof of the above result, and to [11] for the computation
of the Haar stat and representation theory of the compact quantum group
Example 3 : Noncommutative Torus; full quantum isometry group
By similar but somewhat tedious calculations (see [3]) one can also describe
explicitly the full quantum isometry group QISO(A∞,H,D). It is as a
C∗ algebra has eight direct summands, four of which are isomorphic with
the commutative algebra C(T2), and the other four are irrational rotation
algebras.
Theorem 3.3 QISO(Aθ) = ⊕
∗(Uk1, Uk2) (as a C
∗ algebra), where for
odd k, Uk1, Uk2 are the two commuting unitary generators of C(T
2), and for
even k, Uk1Uk2 = exp(4πiθ)Uk2Uk1, i.e. they generate A2θ. The (co)-action
on the generators U, V (say) of Aθ are given by the following :
α0(U) = U⊗(U11+U31)+V⊗(U52+U62)+U
−1⊗(U21+U41)+V
−1⊗(U72+U82),
α0(V ) = U⊗(U51+U71)+V⊗(U12+U22)+U
−1⊗(U61+U81)+V
−1⊗(U32+U42).
From the co-associativity condition, the co-product of QISO(Aθ) can easily
be calculated. For the detailed description of the coproduct, counit, an-
tipode and study of the representation theory of QISO(Aθ), the reader is
referred to [3]. It is interesting to mention here that the quantum isometry
group of Aθ is a Rieffel type deformation of the isometry group (which is
same as the quantum isometry group) of the commutative two-torus. The
commutative two-torus is a subgroup of its isometry group, but when the
isometry group is deformed into QISO(Aθ), the subgroup relation is not
respected, and the deformation of the commutative torus, which is A2θ, sits
in QISO(Aθ) just as a C
∗ subalgebra (in fact a direct summand) but not
as a quantum subgroup any more. This perhaps provides some explanation
of the non-existence of any Hopf algebra structure on the noncommutative
torus.
Acknowledgement : The author would like to thank P. Hajac for draw-
ing his attention to the article [11], and S.L. Woronowicz for many valuable
comments and suggestions which led to substantial improvement of the pa-
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