Papers and preprints
- (with S. Sundar)
Regularity and dimension spectrum of the equivariant spectral triple for the odd dimensional quantum spheres,
arXiv:math/0811.3810.
to appear in Journal of NCG.
Abstract: The odd dimensional quantum sphere $S_q^{2\ell+1}$ is a homogeneous space for the quantum group $SU_q(\ell+1)$. A generic equivariant spectral triple for $S_q^{2\ell+1}$ on its $L_2$ space was constructed by Chakraborty \& Pal in \cite{cha-pal-2008a}. We prove regularity for that spectral triple here. We also compute its dimension spectrum and show that it is simple. We give detailed construction of its smooth function algebra and some related algebras that help proving
regularity and in the computation of the dimension spectrum. Following the idea of Connes for $SU_q(2)$, we first study another spectral triple for $S_q^{2\ell+1}$ equivariant under torus group action constructed by Chakraborty \& Pal in \cite{cha-pal-2007a}. We then derive the results for the $SU_q(\ell+1)$-equivariant triple in the $q=0$ case from those for the torus equivariant triple. For the $q\neq 0$ case, we deduce regularity and dimension spectrum from the $q=0$ case.
- (with P. S. Chakraborty) Characterization of $SU_q(\ell+1)$-equivariant spectral triples for the odd dimensional quantum spheres,
arXiv:math.QA/0701694.
J. Reine Angew. Math. 623 (2008), 25–42.
Abstract: The quantum group SU_q(\ell+1) has a canonical action on the odd dimensional sphere S_q^{2\ell+1}. All odd spectral triples acting on the L_2 space of S_q^{2\ell+1} and equivariant under this action have been characterized. This characterization then leads to the construction of an optimum family of equivariant spectral triples having nontrivial K-homology class. These generalize the results of Chakraborty & Pal for SU_q(2).
- (with P. S. Chakraborty) Torus equivariant spectral triples for odd dimensional quantum spheres coming from $C^*$-extensions,
arXiv:math.KT/0701738.
Letters in Math. Phys. 80 (2007), Number 1, 57-68.
Abstract: The torus group (S^1)^{\ell+1} has a canonical action on the odd dimensional sphere S_q^{2\ell+1}. We take the natural Hilbert space representation where this action is implemented and characterize all odd spectral triples acting on that space and equivariant with respect to that action. This characterization gives a construction of an optimum family of equivariant spectral triples having nontrivial K-homology class thus generalizing our earlier results for SU_q(2). We also relate the triple we construct with the C^*-extension 0\rightarrow \mathcal{K}\otimes C(S^1)\rightarrow C(S_q^{2\ell+3}) \rightarrow C(S_q^{2\ell+1}) \rightarrow 0.
- (with P. S. Chakraborty) Equivariant spectral triples for $SU_q(\ell+1)$ and the odd dimensional quantum spheres,
arXiv:math.QA/0503689.
Abstract: We formulate the notion of equivariance of an operator with respect to a covariant representation of a C^*-dynamical system. We then use a combinatorial technique used by the authors earlier in characterizing spectral triples for SU_q(2) to investigate equivariant spectral triples for two classes of spaces: the quantum groups SU_q(\ell+1) for \ell>1, and the odd dimensional quantum spheres S_q^{2\ell+1} of Vaksman & Soibelman. In the former case, a precise characterization of the sign and the singular values of an equivariant Dirac operator acting on the L_2 space is obtained. Using this, we then exhibit equivariant Dirac operators with nontrivial sign on direct sums of multiple copies of the L_2 space. In the latter case, viewing S_q^{2\ell+1} as a homogeneous space for SU_q(\ell+1), we give a complete characterization of equivariant Dirac operators, and also produce an optimal family of spectral triples with nontrivial K-homology class. - (with P. S. Chakraborty) On equivariant Dirac operators for $SU_q(2)$,
arXiv:math.QA/0501019.
Proc. Indian Acad. Sci. (Math. Sci.) 116(2006), No. 4, 531–541.
We give a decomposition of the spectral triple constructed recently by Dabrowski et al (in math.QA/0411609) in terms of the canonical equivariant spectral triple constructed by us in the paper arxiv:math.KT/0201004 below.
- (with P. S. Chakraborty) Characterization of spectral triples: A combinatorial approach,
arXiv:math.OA/0305157.
Here we persent a new, essentially combinatorial, technique to study Dirac operators on noncommutative spaces. As an illustration, we then look at the groups SU_q(\ell+1) and do a detailed analysis.
- (with P. S. Chakraborty) Equivariant spectral triples and Poincaré duality for $SU_q(2)$,
arXiv:math.OA/0211367.
to appear in Trans. AMS.
Let $\cla$ be the $C^*$-algebra associated with $SU_q(2)$, $\pi$ be the representation by left multiplication on the $L_2$ space of the Haar state and let $D$ be the equivariant Dirac operator for this representation constructed by the authors earlier. We prove in this article that there is no operator other than the scalars in the commutant $\pi(\cla)’$ that has bounded commutator with $D$.
This implies that the equivariant spectral triple under consideration does not admit a rational Poincar\’e dual in the sense of Moscovici, which in particular means that this spectral triple does not extend to a $K$-homology fundamental class for $SU_q(2)$. We also show that a minor modification of this equivariant spectral triple gives a fundamental class and thus implements Poincar\’e duality.
- (with P. S. Chakraborty) Spectral triples and associated Connes-de Rham complex for the quantum $SU(2)$ and the quantum sphere, arxiv:math.QA/0210049
Commun. Math. Phys., 240(2003), No. 3, 447-456.
(The original publication is available at http://www.springerlink.com)
Using techniques developed in our earlier paper (see below), we characterize all spectral triples on SU_q(2) equivariant under the action of S^1\times S^1 and having nontrivial K-homology class. The dimension of such triples can not go below 2. Starting with one such spectral triple, we give a detailed computation of the associated Connes-de Rham cohomology and the space of L_2-forms. At the end, we indicate briefly how to carry out all these constructions for the quantum sphere.
- (with P. S. Chakraborty) Equivariant spectral triples on the quantum SU(2) group,
arxiv:math.KT/0201004
K-Theory, 28(2003), No. 2, 107-126.
An attempt to understand how noncommutative geometry and quantum groups go together. All equivariant finitely summable odd spectral triples for the quantum SU(2) group acting on its L_2-space and having nontrivial pairing with K-theory have been characterized. The dimension of such triples is shown to be 3. The paper also gives first known examples of nontrivial equivariant spectral triples on quantum groups. (for a detailed analysis and a local index formula based on these equivariant triples, see Connes’ paper). As a by-product of the method employed, it is shown that for classical SU(2), there does not exist any p-summable equivariant spectral triple acting on its L_2-space if p\lt 4.
- (with D. Goswami and K. B. Sinha) Stochastic dilation of a quantum dynamical semigroup on a separable unital C*-algebra,
Inf. Dim. Analysis, Quantum Prob. and Related Topics, 3(2000), No. 1, 177-184.
[pdf]
Extends a result of Goswami and Sinha in Von Neumann algebra case to the C*-algebra set up, essentially by making use of Kasparov’s stabilization theorem.
- Regularity of operators on essential extensions of the compacts,
Proc. Amer. Math. Soc, 128(2000), no. 9, 2649-2657.
Continuation of the earlier work (the one below) to a larger class of C*-algebras. Some examples arising naturally in the study of quantum groups are covered.
- Regular operators on Hilbert C*-modules,
J. Operator Theory, 42(1999), 331-350.
Regular operators on a Hilbert module are like closed densely defined operators on a Hilbert space, and one can do a lot of analysis with them. But the problem one faces in many situations with an unbounded operator on a Hilbert module is, is the operator regular in the first place? This paper tries to give an answer in some special cases by translating the problem to that on a C*-algebra.
- On Some Quantum Groups and their Representations,
Ph. D. Thesis, 1995, Indian Statistical Institute.
[pdf]
Essentially consists of results in the papers below, and also a detailed analysis of the regular representation of Eq(2), including a Clebsch-Gordon type decomposition and a treatment of the Haar weight on its dual.
- q-analogues of Graf’s identities from the regular representation of Eq(2), Preprint, 1996.
[pdf]
The identities are not new, but the proof is, which uses quantum groups.
- Haar measure on Eq(2).
Pacific Journal of Mathematics, No. 1, 176(1996), 217-233.
[pdf]
Derives the invariance properties of the Haar weight on the quantum Euclidean group of motions. Unfortunately, I didnt know at the time of writing this that Baaj had already proved all this. Fortunately, however, my proof was entirely different from his. While he used identities involving $q$-functions, mine used operator norm estimates, and in my case one gets lots of identities as by-products.
- A counterexample on idempotent states on compact quantum groups.
Letters in Mathematical Physics, 37(1996), 75-77.
[pdf]
For a compact group, if you take an idempotent measure, there always exists a subgroup such that it is the Haar measure on that subgroup. Here is a counter example in the quantum case, which illustrates that subgroups of quantum groups are scarce in some sense.
- Induced representation and Frobenius reciprocity for compact quantum groups. Proceedings of the Indian Academy of Sciences, No. 2, 105(1995), 157-167.
[pdf]
My first paper; as the title suggests, it extends the classical result to the quantum case.
