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Cluster Representation Theory

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Alexander Goncharov
Yale University/MPIM
Thu, 11/05/2023 - 15:00 - 16:00
MPIM Lecture Hall
Parent event: 

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.

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