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June 19, 2014
Thursday, June 19
Noncommutative Geometry
Time: 13:00
Speaker: Branimir Cacic (Texas A&M)
Title: "A reconstruction theorem for noncommutative G-manifolds"
Room: MC 106

Abstract: 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.