PBW filtrations for quantum groups

Ghislain Fourier (University of Glasgow)

Wednesday 7th October, 2015 16:00-17:00 Maths 522

Abstract

I will recall the PBW filtration for universal enveloping algebras and the famous PBW theorem that tells us the associated graded algebra is a polynomial ring. In the case of quantum groups life is not that simple, the standard degree function and the induced filtration, the quantum group is not even a filtered algebra.

By adjusting the degree functions, I will present a general solution to this problem and show that we can view these admissible degree functions as dimensions of homomorphism spaces (using the Hall algebra realization by Ringel). Finally, we will see that applying these filtration to cyclic modules will provide a monomial basis for any simple finite-dimensional module. These basis were conjectured and constructed before, but with our results we see that are quite natural.

This is a joint work with T.Backhaus, X.Fang (both from Cologne) and M.Reineke (from Wuppertal).

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