Invariance and the twisted Chern character : a case study
arXiv:0704.0111v1  [math.QA]  2 Apr 2007
Invariance and the twisted Chern character : a case study
Debashish Goswami
Stat-Math Unit, Indian Statistical Institute
203, B. T. Road, Kolkata 700108, India.
E-mail : goswamid@isical.ac.in
Abstract
We give details of the proof of the remark made in [7] that the
Chern characters of the canonical generators on the K homology of
the quantum group SUq(2) are not invariant under the natural SUq(2)
coaction. Furthermore, the conjecture made in [7] about the nontriv-
iality of the twisted Chern character coming from an odd equivariant
spectral triple on SUq(2) is settled in the affirmative.
1 Introduction
Noncommutative geometry (NCG) (a la Connes, see [2] ) and the C∗-
algebraic theory of quantum groups (see, for example, [11], [10]) are two
well-developed mathematical areas which share the basic idea of ‘noncom-
mutative mathematics’, namely, to view a general (noncommutative) C∗
algebra as noncommutative analogue of a topological space, equipped with
additional structures resembling and generalizing those in the classical (com-
mutative) situation, e.g. manifold or Lie group structure. A lot of fruitful
interaction between these two areas is thus quite expected. However, such
an interaction was not very common until recently, when a systematic effort
by a number of mathematicians for understanding C∗-algebraic quantum
groups as noncommutative manifolds in the sense of Connes triggered a
rapid and interesting development to this direction. However, quite sur-
prisingly, such an effort was met with a number of obstacles even in the
case of the simplest non-classical quantum group, namely SUq(2) and it was
not so clear for some time whether this (and other standard examples of
quantum groups) could be nicely fitted into the framework of Connes’ NCG
(see [6] and the discussion and references therein). The problem of finding
a nontrivial equivariant spectral triple for SUq(2) was finally settled in the
affirmative in the papers by Chakraborty and Pal ([4], see also [3] and [5]
for subsequent development), which increased the hope for a happy mar-
riage between NCG and quantum group theory. However, even in the case
of SUq(2), a few puzzling questions remain to be answered. One of them
is the issue of invariance of the Chern character, which we have addressed
in [7] and attempted to suggest a solution through the twisted version of
http://arxiv.org/abs/0704.0111v1
the entire cyclic cohomology theory, building on the ideas of [8]. In that
paper, we also made an attempt to study the connection between twisted
and the conventional NCG following a comment in [3]. The present article
is a follow-up of [7], and we mainly concentrate on SUq(2), considering it as
a test-case for comparing the twisted and conventional formulation of NCG.
2 Notation and background
Let A = SUq(2) (with 0 < q < 1) denote the C∗-algebra generated by two
elements α, β satisfying
α∗α+ β∗β = I, αα∗ + q2ββ∗ = I, αβ − qβα = 0,
αβ∗ − qβ∗α = 0, β∗β = ββ∗.
We also denote the ∗-algebra generated by α and β (without taking the
norm completion) by A∞. There is a Hopf ∗ algebra structure on A∞, as
can be seen from, for example, [10]. We denote the canonical coproduct
on A∞ by ∆. We shall also use the so-called Sweedler convention, which
we briefly explain now. For a ∈ A∞, there are finitely many elements
, i = 1, 2, ..., p (say), such that ∆(a) =
⊗a(2)
. For notational
convenience, we abbreviate this as ∆(a) = a(1)⊗a(2). For any positive integer
m, let A∞m be the m-fold algebraic tensor product of A∞. There is a natural
coaction of A∞ on A∞m given by
∆mA(a1 ⊗ a2 ⊗ ...⊗ am) := (a1(1) ⊗ ...am(1))⊗ (a1(2)...am(2)),
using the Sweedler notation, with summation being implied. Let us recall
the convolution ∗ defined in [7]. If φ : A∞m → C is an m-linear functional,
and ψ : A∞ → C is a linear functional, we define their convolution φ ∗ ψ :
A∞m → C by the following :
(φ ∗ ψ)(a1 ⊗ ...⊗ am) := φ(a(1)1 ⊗ ...⊗ a(1)m )ψ(a
1 ...a
using the Sweedler convention. We say that an m-linear functional φ is
invariant if φ ∗ ψ = ψ(1)φ for every functional ψ on A∞.
In [9], the K-homology K∗(A∞) has been explicitly computed. It has
been shown there that K0(A∞) = K1(A∞) = Z, and the Chern charac-
ters (in cyclic cohomology ) of the generators of these K-homology groups,
denoted by [τev] and [τodd] respectively, are also explicitly written down.
3 Main results
3.1 Chern characters are not invariant
In this subsection, we give detailed arguments for a remark made in [7]
about the impossibility of having an invariant Chern character for A∞ under
the conventional (non-twisted) framework of NCG. To make the notion of
invariance precise, we give the following definition (motivated by a comment
by G. Landi, which is gratefully acknowledged).
Definition 3.1 We say that a class [φ] ∈ HCn(A∞) is invariant if there is
an invariant n + 1-linear functional φ′ such that φ′ is a cyclic cocycle and
φ′ ∼ φ (i.e. [φ] = [φ′]).
It is easy to see that the Chern chracter [τev] cannot be invariant. Had it
been so, it would follow from the uniqueness of the Haar state (say h) on
SUq(2) that τev must be a scalar multiple of h. Since τev is a nonzero trace,
it would imply that h is a trace too. But it is known (see [10]) that h is not
a trace.
However, proving that [τodd] is not invariant requires little bit of detailed
arguments. We begin with the following observation.
Lemma 3.2 If τ is a trace on A∞, i.e. τ ∈ HC0(A∞), then we have that
(∂ξ) ∗ τ = ∂(ξ ∗ τ)
for every functional ξ on A∞, where the Hochschild coboundary operator ∂
is defined by
(∂ξ)(a, b) = ξ(ab)− ξ(ba).
Proof :
We shall use the Swedler notation. We have that for a0, a1 ∈ A∞,
(∂ξ ∗ τ)(a0, a1)
= (∂ξ)(a
0 ⊗ a
1 )τ(a
= ξ(a
1 )τ(a
1 )− ξ(a
0 )τ(a
= ξ(a
1 τ(a
1 )− ξ(a
0 )τ(a
0 ) (since τ is a trace)
= (ξ ∗ τ)(a0a1)− (ξ ∗ τ)(a1a0)
= ∂(ξ ∗ τ)(a0, a1).
The above lemma allows us to define the multiplication ∗ at the level
of cohomology classes. More precisely, for [φ] ∈ HC1(A∞) and [η] ∈
HC0(A∞), we set [φ] ∗ [η] := [φ ∗ η] ∈ HC1(A∞), which is well-defined
by the Lemma 3.2. Similarly [η] ∗ [φ] and [η] ∗ [η′] (where [η′] ∈ HC0(A∞))
can be defined. We now recall from [9] that
[τev] ∗ [τev] = [τev], [τev] ∗ [τodd] = [τodd] ∗ [τev] = 0.
We also note that τev(1) = 1 and that τev is a trace, i.e. τev(ab) = τev(ba).
Using this observation, we are now in a position to prove that the Chern
character of the generator of K1(A∞) is not an invariant class.
Theorem 3.3 [τodd] is not invariant.
Proof :
Suppose that there is φ ∼ τodd such that φ is invariant. Then we have
[φ ∗ τev] = [φ] ∗ [τev] = [τodd] ∗ [τev] = 0.
However, since we have φ ∗ τev = τev(1)φ = φ by the invariance of φ, it
follows that [φ] = [φ ∗ τev] = 0, that is, [τodd] = 0, which is a contradiction.
3.2 Nontrivial pairing with the twisted Chern character
As already mentioned in the introduction, in [7] we have made an attempt
to recover the desirable property of invariance by making a departure from
the conventional NCG and using the twisted entire cyclic cohomology. We
briefly recall here some of the basic concepts from that paper and refer the
reader to [7] and the references therein for more details of this approach. We
shall use the results derived in that paper wihout always giving a specific
reference.
Let us give the definition of twisted entire cyclic cohomology for Banach
algebras for simplicity, but note that the theory extends to locally convex
algebras, which we actually need. The extension to the locally convex al-
gebra case follows exactly as remarked in [1, page 370]. So, let A be a
unital Banach algebra, with ‖.‖∗ denoting its Banach norm, and let σ be
a continuous automorphism of A, σ(1) = 1. For n ≥ 0, let Cn be the
space of continuous n + 1-linear functionals φ on A which are σ-invariant,
i.e. φ(σ(a0), ..., σ(an)) = φ(a0, ..., an)∀a0, ..., an ∈ A; and Cn = {0} for
n < 0. We define linear maps Tn, Nn : C
n → Cn, Un : Cn → Cn−1 and
Vn : C
n → Cn+1 by,
(Tnf)(a0, ..., an) = (−1)nf(σ(an), a0, ..., an−1), Nn =
T jn,
(Unf)(a0, ..., an−1) = (−1)nf(a0, ..., an−1, 1),
(Vnf)(a0, ..., an+1) = (−1)n+1f(σ(an+1)a0, a1, ..., an).
Let Bn = Nn−1Un(Tn − I), bn =
j=0 T
n+1 VnT
n. Let B, b be maps on
the complex C ≡ (Cn)n given by B|Cn = Bn, b|Cn = bn. It is easy to
verify (similar to what is done for the untwisted case , e.g. in [2]) that
B2 = 0, b2 = 0 and Bb = −bB, so that we get a bicomplex (Cn,m ≡ Cn−m)
with differentials d1, d2 given by d1 = (n − m + 1)b : Cn,m → Cn+1,m,
: Cn,m → Cn,m+1. Furthermore, let Ce = {(φ2n)n ∈ IN ;φ2n ∈
C2n∀n ∈ IN}, and Co = {(φ2n+1)n ∈ IN ;φ2n+1 ∈ C2n+1∀n ∈ IN}. We
say that an element φ = (φ2n) of C
e is a σ-twisted even entire cochain if
the radius of convergence of the complex power series
‖φ2n‖z
is infinity,
where ‖φ2n‖ := sup‖aj‖∗≤1 |φ2n(a0, ...., a2n)|. Similarly we define σ-twisted
odd entire cochains, and let Ceǫ (A, σ) (Coǫ (A, σ) respectively) denote the set
of σ-twisted even (respectively odd) entire cochains. Let ∂̃ = d1 + d2 , and
we have the short complex Ceǫ (A, σ)
Coǫ (A, σ). We call the cohomology
of this complex the σ-twisted entire cyclic cohomology of A and denote it
by H∗ǫ (A, σ). Let Aσ = {a ∈ A : σ(a) = a} be the fixed point subalgebra
for the automorphism σ. There is a canonical pairing < ., . >σ,ǫ: K∗(Aσ)×
H∗ǫ (A, σ) → C. We shall need the pairing for the odd case, which we write
down :
< [u], [ψ] >≡< [u], [ψ] >σ,ǫ=
(−1)n n!
(2n + 1)!
ψ2n+1(u
−1, u, ..., u−1, u),
where [u] ∈ K1(Aσ) and [ψ] ∈ H1ǫ (A, σ).
Definition 3.4 Let H be a separable Hilbert space, A∞ be a ∗ subalgebra
(not necessarily complete) of B(H), R be a positive (possibly unbounded)
operator on H, D be a self-adjoint operator in H with compact resolvents
such that the following hold :
(i) [D, a] ∈ B(H) ∀a ∈ A∞,
(ii) R commutes with D,
(iii) For any real number s and a ∈ A∞, σs(a) := R−saRs is bounded and be-
longs to A∞. Furthermore, for any positive integer n, sups∈[−n,n] ‖σs(a)‖ <
Then we call the quadruple (A∞,H,D,R) an odd R-twisted spectral data.
We say that the odd twisted spectral data is Θ-summable if Re−tD
is trace-
class for all t > 0.
Let us now recall the construction of twisted Chern character from a
given odd twisted spectral data (A∞,H,D,R). Let B denote the set of
all A ∈ B(H) for which σs(A) := R−sARs ∈ B(H) for all real number s,
[D,A] ∈ B(H) and s 7→ ‖σs(A)‖ is bounded over compact subsets of the real
line. In particular, A∞ ⊆ B. We define for n ∈ IN an n+1-linear functional
Fn on B by the formula
Fn(A0, ..., An) =
Tr(A0e
...Ane
R)dt0...dtn,
where Σn = {(t0, ..., tn) : ti ≥ 0,
i=0 ti = 1}.
Let us now equip A∞ with the locally convex topology given by the fam-
ily of Banach norms ‖.‖∗,n, n = 1, 2, ..., where ‖a‖∗,n := sups∈[−n,n](‖σs(a)‖+
‖[D,σs(a)]‖). Let A denote the completion of A∞ under this topology, and
thus A is Frechet space. We can construct the (twisted) Chern character in
Hoǫ (A, σ), where σ = σ1, which extends on the whole of A by continuity.
Theorem 3.5 Let φo ≡ (φ2n+1)n be defined by
φ2n+1(a0, ..., a2n+1) =
2iF2n+1(a0, [D, a1], ..., [D, a2n+1]), ai ∈ A.
Then we have (b+B)φo = 0, hence ψo ≡ ((2n + 1)!φ2n+1)n ∈ Hoǫ (A, σ).
We shall also need some results from the theory of semifinite spectral
triples and the corresponding JLO cocycles and index formula, as discussed
in, for example, [1]. An odd semifinite spectral triple is given by (C,N ,K,D),
where K is a separable Hilbert space, N ⊆ B(K) is a von Neumann algebra
with a faithful semifinite normal trace (say τ), D is a self-adjoint operator
affiliated to N , C is a ∗-subalgebra of B(K) such that [D, c] ∈ B(K) for all
c ∈ C. In the terminology of [1], (N ,D) is also called an odd, unbounded
Breuer-Fredholm module for the norm-closure of C. It is called Θ-summable
if τ(e−tD
) < ∞ for all t > 0. For a Θ-summable semifinite spectral triple,
there is a canonical construction of JLO cocycle and index theorem (see [1]),
which are very similar to their counterparts in the conventional framework
of NCG.
Let us now settle in the affirmative conjecture made in [7] about the
nontriviality of the twisted Chern character of a natural twisted spectral data
obtained from the equivariant spectral triple of [4]. For reader’s convenience,
we briefly recall the construction of this equivariant spectral triple. Let us
index the space of irreducible (co-)representations of SUq(2) by half-integers,
i.e. n = 0, 1
, 1, ...; and index the orthonormal basis of the corresponding
(2n + 1)2 dimensional subspace of L2(SUq(2), h) by i, j = −n, ..., n, instead
of 1, 2, ..., (2n + 1). Thus, let us consider the orthonormal basis eni,j , n =
, ...; i, j = −n,−n + 1, ..., n in the notation of [4]. We consider any of
the equivariant spectral triples constructed by the authors of [4] and in the
associated Hilbert space H = L2(SUq(2), h) define the following positive
unbounded operator R :
R(eni,j) = q
−2i−2jeni,j ,
n = 0, 1
, , 1, ...; i, j = −n,−n+ 1, ..., n. Let us choose a spectral triple given
by the Dirac operator D on H, defined by
D(eni,j) = d(n, i)e
i,j ,
where d(n, i) are as in (3.12) of [4], i.e. d(n, i) = 2n + 1 if n = i, d(n, i) =
−(2n+ 1) otherwise. It can easily be seen that (A∞,H,D,R) is an odd R-
twisted spectral data and furthermore, the fixed point subalgebra SUq(2)σ
for σ(.) = R−1 ·R is the unital ∗-algebra generated by β, so it contains u =
∗β)(β−I)+I which can be chosen to be a generator of K1(SUq(2)) = Z
(see [4]). It is easily seen that the map from K1(C
∗(u)) to K1(SUq(2)),
induced by the inclusion map, is an isomorphism of the K1-groups (where
C∗(u) denotes the unital C∗-algebra generated by u). Thus, we can consider
the pairing of the twisted Chern character with K1(C
∗(u)), and in turn with
K1(SUq(2)) using the isomorphism noted before. The important question
raised in [7] is whether we recover the nontrivial pairing obtained in [4] in
our twisted framework, and in what follows, we shall give an affirmative
answer to this question.
Theorem 3.6 The pairing between K1(SUq(2)σ) ∼= K1(SUq(2)) and the
(twisted) Chern character of the above twisted spectral data coincides with
the pairing between K1(SUq(2)) and the Chern character of the (non-twisted)
spectral triple (A∞,H,D). In particular, this pairing is nontrivial.
Proof :
Let N be the von Neumann algebra in B(H) generated by β and f(D) for
all bounded measurable functions f : R → C. Since R commutes with both
β and D, it is easy to see that the functional N ∋ X 7→ τ(X) := Tr(XR)
defines a faithful, normal, semifinite trace on the von Neumann algebra N .
Moreover, (N ,D) is an unbounded Θ-summable Breuer-Fredholm module
for the norm-closure of the unital ∗-algebra (say C) generated by β.
Moreover, it follows from the fact that R commutes with D and u that
the pairing of [u] with the twisted Chern character (say ψo ≡ (ψ2n+1))
coming from the twisted spectral data (A∞,H,D,R) is given by
< [u], [ψo] >
(−1)n n!
(2n+ 1)!
ψ2n+1(u
−1, u, ..., u−1, u)
(−1)nn!
Σ2n+1
Tr(u−1e−t0D
[D,u]et1D
...[D,u]et2n+1D
R)dt0...dt2n+1,
(−1)nn!
Σ2n+1
τ(u−1e−t0D
[D,u]et1D
...[D,u]et2n+1D
)dt0...dt2n+1
which is nothing but the pairing between [u] ∈ K1(C) and the Breuer-
Fredholm module (N ,D) mentioned before. By Theorem 10.8 of [1] and
a straightforward but somewhat lengthy calculation along the lines of index
computation in [4], we can show that the value of this pairing is equal to
−indτ (A) ≡ −(τ(PA) − τ(QA)) for the following operator A : H0 → H0,
where H0 is the closed subspace spanned by {enn,j , n = 0, 12 , ..., j = −n,−n+
1, ..., n}, PA, QA are the orthogonal projections onto the kernel of A and
the kernel of A∗ respectively and where r is a positive integer such that
q2r < 1
< q2r−2 :
Aenn,j = −q(n+j)(2r+1)(1−q2(n−j))r(1−q2(n−j−1))
,j− 1
+(1−q2r(n+j)(1−q2(n−j))r)enn,j.
It can be verified by computations as in [4] that Ker(A) = {0} and Ker(A∗)
is the one dimensional subspace spanned by the vector ξ =
n,−n,
where p 1
= 1 and for n ≥ 3
1− (1− q4n−2)r
(1− q4n) 12 (1− q4n−2)r
1− (1− q2)r
(1 − q4) 12 (1− q2)r
Clearly, since Ren−n,n = e
n,−n, we have Rξ = ξ and thus
−indτ (A) =
‖ξ‖2 τ(|ξ >< ξ|) =
‖ξ‖2Tr(R|ξ >< ξ|) = 1,
which is the same as the value of the pairing between [u] ∈ K1(SUq(2))
and the conventional Chern character corresponding to the spectral triple
constructed in [4]. ✷
Thus we see that both the conventional and twisted frameworks of NCG give
essentially the same results for the example we considered, namely SUq(2).
The aparent weakness of the twisted NCG arising from the fact that the
twisted cyclic cohomology can be paired naturally with only the K theory
of the invariant subalgebra and not of the whole algebra, does not seem
to pose any essential difficulty for studying the noncommutative geometric
aspects of SUq(2), since by a suitable choice of the twisting operator R
as we did one could make sure that the K theory of the corresponding
invariant subalgebra is isomorphic with the K theory of the whole, and also
the pairing between the Chern character and the generator of the K theory
in the twisted framework is equal to the similar pairing in the ordinary
(non-twisted) framework of NCG. It will be important and interesting to
investigate whether a similar fact remains true for a larger class of quantum
groups, and we hope to pursue this in the future.
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