Quantum groups via finite-dimensional algebras


Stephen Doty


Loyola University Chicago


Tue, 10/04/2018 - 12:00pm to 1:00pm


RC-4082, The Red Centre, UNSW


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


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

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