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

Axioms for nonunital spectral triples, extending those introduced in the unital case by Connes, are proposed. As a guide, and for the sake of their importance in noncommutative quantum field theory, the spaces R2N endowed with Moyal products are intensively investigated. Some physical applications, such as the construction of noncommutative Wick monomials and the computation of the Connes–Lott functional action, are given for these noncommutative hyperplanes.

Moyal Planes are Spectral Triples

http://library.um.ac.id/free-contents/downloadpdf.php/buku/methods-of-modern-mathematical-physics-vol-i-functional-analysis-reed-michael-barry-simon-12066.pdf

Methods of modern mathematical physics (vol.) I : functional analysis / Reed Michael, Barry Simon)

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

An Introduction to Noncommutative Differential Geometry and its Physical Applications: Extensions of Space-Time

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

Geometry on the quantum Heisenberg manifold

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

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.

/pdf/probability-and-geometry-on-some-noncommutative-manifolds-5083rwv8zk.pdf

Partha Sarathi Chakraborty

Papers

Equivariant spectral triples on the quantum SU(2) group

Geometry on the quantum Heisenberg manifold

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

Probability and geometry on some noncommutative manifolds