Quantum groups. Formalism and applications

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We review the definition and elementary properties of the Crainic-Moerdijk homology in the setting of ample groupoids. We show that there is a natural homomorphism from the homology of a higher rank graph to that of its path groupoid. In joint work my coauthors, Carla Farsi, David Pask and Aidan Sims, and I have shown that the isomorphism in the conjecture holds for the path groupoids of higher rank graphs of ranks one or two even when the unit space is no longer compact. Abelian operator algebras are understood.

General operator algebras can be analyzed by examination of large abelian subalgebras. Renault's seminal result relates Cartan subalgebras, which are abelian and particularly nice, to algebraic structures known as groupoids. While the formula for the lowest index of an elliptic operator D on a closed manifold M which coincides with its Fredholm index reproduces the Atiyah-Singer index theorem, our formula for the highest index of D associated to a volume cocycle yields an extension to arbitrary manifolds of any dimension of the Helton-Howe formula for the trace of multicommutators of classical Toeplitz operators on odd-dimensional spheres.

Center for Computational Quantum Physics

In fact, the totality of higher analytic indices for an elliptic operator D amounts to a representation of the Connes-Chern character of the K-homology cycle determined by D in terms of expressions which extrapolate the Helton-Howe formula below the dimension of M. Based on joint work with H. Let G be a semisimple real Lie group with simply connected complexification G c.

Let U be the maximal compact subgroup of G c. Then U is a compact semisimple Lie group which admits a q -deformation into a compact quantum group U q.

Heat kernel expansion in the background field formalism - Scholarpedia

Letzter through an adaptation of the universal K-matrix formalism due to M. Balagovic and S. This is joint work with Marco Matassa.

This new episode in noncommutative differential geometry started when a Gauss-Bonnet-type theorem was proved by Connes and Tretkoff for a curved noncommutative two torus. In this talk, I shall try to highlight what has happened in the subject by giving an account of the most recent developments. On a closed i. Thus, on odd-dimensional manifolds, interesting examples can only be obtained by dropping either "elliptic" or "differential". This talk will explain the two cases.

If "differential" is dropped, then the examples are Toeplitz operators which are elliptic pseudo-differential operators of order zero. A corollary is the result of Louis Boutet de Monvel about Toeplitz operators associated to the boundary of a strictly pseudo-convex domain. If "elliptic" is dropped, then examples are the differential operators on contact manifolds studied by E. The connection between Hochschild and cyclic cohomologies with generalized de Rham homology and index theories for arbitrary algebras has long been established by the work of Connes, Karoubi, Loday, Feigin, Tsygan, et al.

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Group Theory in Quantum Mechanics

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