A new approach to the $ı$Serre and Serre-Lusztig relations for $ı$quantum groups

Abstract

We give a new, conceptual proof of the $ı$Serre and Serre-Lusztig relations for $ı$quantum groups. The key to our approach is a new formula for the comultiplication of the $ı$-divided powers, which allows us to reformulate the relations in terms of the adjoint action. We then obtain a proof using properties of the adjoint representation. The flexibility of this approach allows us to establish a more general family of relations which seem difficult to establish otherwise.

arXiv comment13 pages. v2, completely rewritten, new title, self-contained new proofs of iSerre and Serre-Lusztig relations added