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Papers and preprints

  1. (with P. S. Chakraborty) Torus equivariant spectral triples for odd dimensional quantum spheres coming from $C^*$-extensions,
    arXiv:math.KT/0701738.
    To appear in Letters in Math. Phys.
    (The original publication is available at http://www.springerlink.com)

    Generalization of the results in the paper math.QA/0210049.

  2. (with P. S. Chakraborty) Characterization of $SU_q(\ell+1)$-equivariant spectral triples for the odd dimensional quantum spheres,
    arXiv:math.QA/0701694.
    (Submitted)

    This is the result of a bit of reorganization of the material in the paper below. Since the results for odd dimensional spheres are complete, we decided to take it out and write a separate article.

  3. (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.

  4. (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.

  5. (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.

  6. (with P. S. Chakraborty) Remark on Poincare duality for $SU_q(2)$,
    arXiv:math.OA/0211367.

    We show that Poincare duality fails for the equivariant spectral triple on the quantum SU(2) group.

  7. (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.

  8. (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<4. This shows that the classical Dirac operator for SU(2), which resides on two copies of L_2(SU(2)), is in some sense, minimal.

  9. (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] [ps]

    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.

  10. 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.

  11. 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.

  12. On Some Quantum Groups and their Representations,
    Ph. D. Thesis, 1995, Indian Statistical Institute.
    [pdf] [ps]

    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.

  13. q-analogues of Graf's identities from the regular representation of Eq(2),
    Preprint, 1996.
    [pdf] [ps]

    The identities are not new, but the proof is, which uses quantum groups.

  14. Haar measure on Eq(2).
    Pacific Journal of Mathematics, No. 1, 176(1996), 217-233.
    [pdf] [ps]

    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.

  15. A counterexample on idempotent states on compact quantum groups.
    Letters in Mathematical Physics, 37(1996), 75-77.
    [pdf] [ps]

    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.

  16. Induced representation and Frobenius reciprocity for compact quantum groups.
    Proceedings of the Indian Academy of Sciences, No. 2, 105(1995), 157-167.
    [pdf] [ps]

    My first paper; as the title suggests, it extends the classical result to the quantum case.


Last modified: Wed July 5 11:27:38 IST 2006