By Vyjayanthi Chari and Andrew Pressley: 651 pp., £22.95 (US$34.95), isbn 0 521 55884 0 (Cambridge University Press, 1994).

A guide to quantum groups

This is the introduction and bibliography for lecture notes of a course given at the Summer School on Noncommutative Geometry and Applications, sponsored by the European Mathematical Society, at Monsaraz and Lisboa, Portugal, September 1-10, 1997 In the published version, an epilogue of recent developments and many new references from 1998-2006 have been added 
1 Commutative geometry from the noncommutative point of view 
2 Spectral triples on the Riemann sphere 
3 Real spectral triples, the axiomatic foundation 
4 Geometries on the noncommutative torus 
5 The noncommutative integral 
6 Quantization and the tangent groupoid 
7 Equivalence of geometries 
8 Action functionals 
9 Epilogue: new directions

An Introduction to Noncommutative Geometry

We construct a 3+-summable spectral triple Open image in new window over the quantum group SUq(2) which is equivariant with respect to a left and a right action of Open image in new window The geometry is isospectral to the classical case since the spectrum of the operator D is the same as that of the usual Dirac operator on the 3-dimensional round sphere. The presence of an equivariant real structure J demands a modification in the axiomatic framework of spectral geometry, whereby the commutant and first-order properties need be satisfied only modulo infinitesimals of arbitrary high order.

The Dirac Operator on SU q (2)

We analyse the non-commutative space underlying the quantum group -symmetry. We shall explain how this naturally leads to the general concept of invariant cyclic cohomology in the framework of quantum group symmetries.AMS 2000 Mathematics subject classification: Primary 81R50; 19K33; 46L; 58B34

CYCLIC COHOMOLOGY, QUANTUM GROUP SYMMETRIES AND THE LOCAL INDEX FORMULA FOR SU q (2)

The free analogues of $U(n)$ in Woronowicz' theory \cite{w2} are the compact matrix quantum groups $\{ A_u(F)|F\in GL(n,\mathbb C)\}$ introduced by Wang and Van Daele. We classify here their irreducible representations. Their fusion rules turn to be related to the combinatorics of Voiculescu's circular variable. If $F\bar{F}\in\mathbb R I_n$ we find an embedding $A_u(F)_{red}\hookrightarrow C(\mathbb T)*_{red}A_o(F)$, where $A_o(F)$ is the deformation of $SU(2)$ studied in \cite{ban}. We use the representation theory and Powers' method for showing that the reduced algebras $A_u(F)_{red}$ are simple, with at most one trace.

Le groupe quantique compact libre U(n)

We characterize all equivariant odd spectral triples for the quantum SU(2) group acting on its L2-space 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.

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Equivariant spectral triples on the quantum SU(2) group

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.

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Spectral triples and associated Connes-de Rham complex for the quantum SU(2) and the quantum sphere

A simple example is given to illustrate that an idempotent state may not be the Haar state of any subgroup in the case of compact quantum groups.

A counterexample on idempotent states on a compact quantum group

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 $SU_q(2)$.

Characterization of $SU_q(\ell+1)$-equivariant spectral triples for the odd dimensional quantum spheres

The quantum group SUq(l + 1) has a canonical action on the odd dimensional sphere S 2l+1 q . All odd spectral triples acting on the L2 space of S 2l+1 q 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 SUq(2). AMS Subject Classification No.: 58B34, 46L87, 19K33

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Arupkumar Pal

Papers

Equivariant spectral triples on the quantum SU(2) group

Spectral triples and associated Connes-de Rham complex for the quantum SU(2) and the quantum sphere

A counterexample on idempotent states on a compact quantum group

Characterization of $SU_q(\ell+1)$-equivariant spectral triples for the odd dimensional quantum spheres

Characterization of SU q (ℓ + 1)-equivariant spectral triples for the odd dimensional quantum spheres