Actions of monoidal categories and representations of Cartan type Lie algebras

TL;DR

This paper explores how monoidal categories act on representations of Lie-Rinehart and Leibniz pairs, introducing crossed homomorphisms to unify and extend existing Lie algebra representation theories, including new results for generalized Witt algebras.

Contribution

It introduces a new bifunctor based on crossed homomorphisms that unifies and generalizes various Lie algebra representation constructions.

Findings

Established monoidal category actions on Lie-Rinehart and Leibniz pair representations

Constructed new weak and admissible representations for Lie-Rinehart algebras

Revealed new representations for generalized Witt algebras

Abstract

Using crossed homomorphisms, we show that the category of weak representations (resp. admissible representations) of Lie-Rinehart algebras (resp. Leibniz pairs) is a left module category over the monoidal category of representations of Lie algebras. In particular, the corresponding bifunctor of monoidal categories is established to give new weak representations (resp. admissible representations) of Lie-Rinehart algebras (resp. Leibniz pairs). This generalizes and unifies various existing constructions of representations of many Lie algebras by using this new bifunctor. We construct some crossed homomorphisms in different situations and use our actions of monoidal categories to recover some known constructions of representations of various Lie algebras, also to obtain new representations for generalized Witt algebras and their Lie subalgebras. The cohomology theory of crossed homomorphisms between Lie algebras is introduced and used to study linear deformations of crossed homomorphisms.