XXXVIII Workshop on Geometric Methods in Physics 30.06-6.07.2019
VIII School on Geometry and Physics 24-28.06.2019

Sergiy Neshveyev

Quantization of subgroups of the affine group

Although the problem of quantization of Lie bialgebras in the purely algebraic (formal) setting was solved in full generality in the 1990s by Etingof and Kazhdan, the list of noncompact Lie bialgebras admitting a quantization in the analytic (operator algebraic) setting is still quite short. In my talk I will present a quantization procedure for a class of semidirect product groups $G$ defined by an action of a locally compact group $Q$ on a locally compact abelian group $V$. In fact, three equivalent procedures will be given. The first is via reflection across the Hopf-Galois object defined by a distinguished irreducible representation of $G$. The second is via cocycle twisting related to the Kohn-Nirenberg quantization. The third is via a bicrossed product defined by a pair of groups isomorphic to $Q$. In the simplest case of the $ax+b$ group over the reals the equivalence of the second and the third constructions imply that certain quantum groups defined by Baaj-Skandalis and Stachura are indeed isomorphic. I will also show that the first construction makes sense in greater generality. Specifically, every locally compact group with group von Neumann algebra isomorphic to the algebra of bounded operators on a Hilbert space admits a canonical quantization which is neither commutative nor cocommutative. (Joint work with Pierre Bieliavsky, Victor Gayral and Lars Tuset.)

Event sponsored by:
of Bialystok
University of Bialystok

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