Published/Accepted

  1. A Covariant Quantum Stochastic Dilation Theory (with D.Goswami and K. B. Sinha), Stochastics in finite and infinite dimensions, Trends Math., Birkhäuser Boston, 89-99 (2001).
  2. Probability and Geometry on some Noncommutative Manifolds (with D.Goswami and K. B. Sinha), Journal of Operator Theory , 49, No 1, 187-203, 2003.
    Abstract: In a noncommutative torus, effect of perturbation by inner derivation on the associated quantum stochastic process and geometric parameters like volume and scalar curvature have been studied. Cohomological calculations show that the above perturbation produces new spectral triples. Also for the Weyl C*-algebra, the Laplacian associated with a natural stochastic process is obtained and associated volume form is calculated.
  3. Geometry on the Quantum Heisenberg Manifolds, ( with K. B. Sinha) Journal of Functional Analysis, 203 , no 2, 2003, 425-452.
    Abstract: A class of C*algebras called quantum Heisenberg manifolds were introduced by Rieffel in (Comm. Math. Phys. 122 (1989) 531) as strict deformation quantization of Heisenberg manifolds. Using the ergodic action of Heisenberg group we construct a family of spectral triples. It is shown that associatedKasparov modules are homotopic. We also show that they induce cohomologous elements in entire cyclic cohomology. The space of Connes-deRham forms have been explicitly calculated. Then we characterize torsionless/unitary connections andshow that there does not exist a connection that is simultaneously torsionless and unitary. Explicit examples of connections are produced with negative scalar curvature. This part illustrates computations involving some of the concepts introduced in Frohlich et al. (Comm. Math. Phys. 203 (1999) 119), for which to the best of our knowledge no infinite-dimensional example is known other that the noncommutative torus.
  4. Equivariant Spectral Triples on the Quantum SU(2) Group ( with A. Pal), K-theory, 28 , No 2, 2003, 107-126.
    Abstract: We characterize all equivariant odd spectral triples for the quantum SU(2) group acting on its L2space and having a nontrivial Chern character. It is shown that the dimension of an equivariant spectral triple is at least three, and given any element of the K-homology group of SUq(2), there is an equivariant odd spectral triple of dimension 3 inducing that element. The method employed to get equivariant spectral triples in the quantum case is then used for classical SU(2), and we prove that for p < 4, there does not exist any equivariant spectral triple with nontrivial K-homology class and dimension p acting on the L2-space.
  5. Spectral Triples and Associated Calculus on the Quantum SU(2) Group and Podles Sphere (with A. Pal), Communications in Mathematical Physics, 240, No 3, 2003, 447-456.
    Abstract: In this article, we construct spectral triples for the C*-algebra of continuous functions on the quantum SU(2) group and the quantum sphere. There have been various approaches towards building a calculus on quantum spaces, but there seem to be very few instances of computations outlined in Chapter 6, [5].We give detailed computations of the associated Connes-de Rham complex and the space of L2-forms.
  6. Invariant measure and a limit theorem for some generalized Gauss maps (with A. Dasgupta), Journal of Theoretical Probability, 17, no 2, 2004, 387-401.
    Abstract: Continued fractions w.r.t. a specified class of numbers is considered. The invariant measures of the corresponding transformations are identified connecting the continued fractions with geodesics on the upper half plane. A problem of convergence in distribution of sums of the coefficients of the continued fraction is also considered.
  7. Equivariant Spectral Triples for SUq(n), n>2 : nonexistence results, 2355-2356, Oberwolfach Reports, No 45, 2004.
  8. Metrics on the Quantum Heisenberg Manifolds. Journal of Operator Theory, 54 (2005), no. 1, 93--100.
    Abstract: Recently Rieffel has introduced the notion of compact quantum metric spaces. He has produced examples using ergodic action of compact Lie groups. In the present note we construct examples of compact quantum metric spaces on the quantum Heisenberg manifolds using ergodic action of the Heisenberg group.
  9. On equivariant Dirac operators for SUq(2), (With A. Pal), Proceedings of Indian Academy of Sciences 116(2006), no. 4, 531-541.
    Abstract: We explain the notion of minimality for an equivariant spectral triple and show that the triple for the quantum SU(2) group constructed by Chakraborty and Pal in [4, above] is minimal. We also give a decomposition of the spectral triple constructed by Dabrowski et al in terms of the minimal triple constructed in [2].
  10. Torus equivariant spectral triples for odd-dimensional quantum spheres coming from C*-extensions, (With A. Pal), Lett. Math. Phys. 80(2007), no. 1, 57--68.
    Abstract: The torus group (S1)(l+1) has a canonical action on the odd-dimensional sphere Sq(2l+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 SUq(2). We also relate the triple we construct with the C*-extension 0---›K ⊗ C(S1)---› K ⊗C(Sq2l+3)---›C(Sq2l+1)---›0.
  11. Characterization of SUq(l+1)$-equivariant spectral triples for the odd dimensional quantum spheres (With A. Pal), Journal für die reine und angewandte Mathematik. 623 (2008), 25-42
    Abstract: The quantum group SUq(l+1) has a canonical action on the odd dimensional sphere Sq(2l+1). All odd spectral triples acting on the L2 space of Sq(2l+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 and Pal for SUq(2).
  12. The geometry of determinant line bundles in noncommutative geometry, (With Mathai Varghese), Journal of Noncommutative Geometry 3 (2009), 559-578.
    Abstract: This article is concerned with the study of the geometry of determinant line bundles associated to families of spectral triples parametrized by the moduli space of gauge equivalence classes of Hermitian connections on a Hermitian finite projective module. We illustrate our results with some examples that arise in noncommutative geometry.
  13. EQUIVARIANT SPECTRAL TRIPLES AND POINCARÉ DUALITY FOR SUq(2) (With A. Pal),TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY. 362, Number 8, 2010, 4099-4115.
    Abstract: Let A be the C*-algebra associated with SUq(2), let π be the representation by left multiplication on the L2 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 π(A)' that has bounded commutator with D. This implies that the equivariant spectral triple under consideration does not admit a rational Poincaré dual in the sense of Moscovici, which in particular means that this spectral triple does not extend to a K-homology fundamental class for SUq(2). We also show that a minor modification of this equivariant spectral triple gives a fundamental class and thus implements Poincaré duality.
  14. Quantum double suspension and spectral triples (With S. Sundar), Journal of Functional Analysis, 260 , Number 9, 2011, 2716-2741.
    Abstract:In this paper we are concerned with the construction of a general principle that will allow us to produce regular spectral triples with finite and simple dimension spectrum. We introduce the notion of weak heat kernel asymptotic expansion (WHKAE) property of a spectral triple and show that the weak heat kernel asymptotic expansion allows one to conclude that the spectral triple is regular with finite simple dimension spectrum. The usual heat kernel expansion implies this property. The notion of quantum double suspension of a C*-algebra was introduced by Hong and Szymanski. Here we introduce the quantum double suspension of a spectral triple and show that the WHKAE is stable under quantum double suspension. Therefore quantum double suspending compact Riemannian spin manifolds iteratively we get many examples of regular spectral triples with finite simple dimension spectrum. This covers all the odd-dimensional quantum spheres. Our methods also apply to the case of noncommutative torus.
  15. From C*-algebra Extensions to Compact Quantum Metric Spaces, Quantum SU(2), Podleś Spheres and Other Examples, Jour. Australian Math. Soc., 90 , Issue 1, 2011, 1-8.
    Abstract:We construct compact quantum metric spaces starting from a C*-algebra extension with a positive splitting. As special cases, we discuss Toeplitz algebras, quantum SU(2) and Podleś spheres.
  16. K-groups of the Quantum Homogeneous Space SUq(n)/SUq(n-2), (With S. Sundar), Pacific Journal of Mathematics, 252 , Number 2, 2011, 275-292.
    Abstract:Quantum Stiefel manifolds were introduced by Vainerman and Podkolzin, who classified the irreducible representations of the C*-algebras underlying such manifolds. We compute the K-groups of the quantum homogeneous spaces SUq(n)/SUq(n - 2) for n ≥ 3. In the case n = 3, we show that K1 is a free ℤ-module, and the fundamental unitary for quantum SU(3) is part of a basis for K1.
Most of these are available at the ARXIV.