Noncommutative Geometry

Speaker: Branimir Cacic (Texas A&M)

"An introduction to strict deformation quantisation"

Time: 14:30

Room: MC 106

In deformation quantisation, one passes from a classical system to a quantum system by deforming the commutative algebra of classical observables into a noncommutative algebra of quantum observables in a manner consistent with the correspondence principle. In the presence of a smooth action of a compact abelian Lie group $G$ (e.g., a torus), one can rigorously effect deformation quantisation through a method, due to Rieffel, called strict deformation quantisation. In this talk, I'll give an introduction to strict deformation quantisation of Frechet $G$-pre-$C^\ast$-algebras, together with the corresponding deformation of $G$-equivariant finitely generated projective modules and $\ast$-representations. If time permits, I'll also introduce Connes--Landi deformation, which, as was observed by Sitarz and Varilly, is precisely strict deformation quantisation for $G$-equivariant spectral triples.

Noncommutative Geometry

Speaker: Branimir Cacic (Texas A&M)

"A reconstruction theorem for noncommutative G-manifolds"

Time: 13:00

Room: MC 106

Just as one can construct the noncommutative $2$-torus as the strict deformation quantisation of the commutative $2$-torus along the translation action, so too can one more generally construct noncommutative $G$-manifolds, namely, strict deformation quantisations of commutative spectral triples along the action of a compact abelian Lie group $G$. I will propose an abstract definition of noncommutative $G$-manifold, analogous to the definition of commutative spectral triple, and show that the deformation of an abstract noncommutative $G$-manifold with deformation parameter $\theta \in H^2(\hat{G},\mathbb{T})$ by a class $\theta^\prime \in H^2(\hat{G},\mathbb{T})$ yields an abstract noncommutative $G$-manifold with deformation parameter $\theta+\theta^\prime$; combined with Connes's reconstruction theorem for commutative spectral triples, this yields the analogue of Connes's reconstruction theorem for noncommutative $G$-manifolds. If time permits, I will also discuss a Pontrjagin-dual version of the Connes--Dubois-Violette splitting homomorphism and use it to show that sufficiently well-behaved rational noncommutative $\mathbb{T}^N$-manifolds are, in fact, almost-commutative.