Quantum groups via finite-dimensional algebras

Abstract:

The Drinfeld-Jimbo definition of a quantised enveloping algebra by generators and relations is a qq-analogue of Serre's presentation of a semisimple Lie algebra. The most complicated relations in the presentations are sometimes called the "Serre relations" and "qq-Serre relations".

For a given finite saturated set of weights, one can truncate the quantised enveloping algebra by passing to a finite-dimensional quotient algebra which "sees" only the representations with weights belonging to the set. The resulting quotient algebras are called generalised qq-Schur algebras; the famed qq-Schur algebras in type A introduced by Dipper and James form just one class of examples. In 2003 I gave a presentation by generators and relations of generalised q-Schur algebras. This was motivated by earlier joint work with Giaquinto, and related work by Du and Parshall, which obtained presentations of the Dipper-James qq-Schur algebras.

All of those presentations were flawed, in that they imposed the q-Serre relations uneccessarily. I will explain why the qq-Serre relations follow from the other defining relations, and discuss the simplified presentations. Analogous statements hold in the classical (q=1q=1) case.