Speaker:
Alexander Goncharov
Affiliation:
Yale University/MPIM
Date:
Thu, 11/05/2023 - 15:00 - 16:00
Let S be an oriented surface with a finite collection of points on the boundary, and G any split reductive group (with connected center).
Then there is a moduli space P(G,S) parametrizing G-local systems on S equipped with certain boundary data.
It carries a canonical cluster Poisson structure, equivariant under the action of a large discrete group, containing the mapping class group of S.
Therefore the cluster quantization construction, developed by V. Fock and myself, assigns to the pair (G,S) a non-commutative *-algebra A(G,S;h)
together with its principal series of (infinite dimensional) *-representations. The assignment S --> *-representations of A(G,S;h)
should provide a continuous version of the modular functor.
The representation theory of quantum groups becomes a part of the representation theory of these algebras.
Its new feature is that both objects and Hom's between them are representations of the algebras A(G,S;h).
The talk is based on the joint work with Linhui Shen.
