Combinatorics of monoidal actions in Lie-algebraic context

Abstract

This paper is, essentially, a survey related to the problem of understanding the combinatorics of the action of the monoidal category of finite dimensional modules over a simple finite dimensional Lie algebra on various categories of Lie algebra modules. A special attention is payed to the Lie algebras ${\mathfrak{sl}}_2$ and ${\mathfrak{sl}}_3$. A few new general results are collected at the end.

Submitted to a conference proceedings