Combinatorics of monoidal actions in Lie-algebraic context

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 ${\mathfrak{s}\mathfrak{l}}_2$ and ${\mathfrak{s}\mathfrak{l}}_3$. A few new general results are collected at the end.