Cohomology of morphism Lie algebras and some applications

TL;DR

This paper introduces a cohomology theory for morphism Lie algebras, explores their deformations and extensions, and relates their cohomology to that of morphism Lie groups, advancing understanding of their algebraic and geometric structures.

Contribution

It defines cohomology for morphism Lie algebras and applies it to study deformations, extensions, and classifications, also connecting algebraic and group cohomology.

Findings

Cohomology of morphism Lie algebras is well-defined and useful.

Deformations and extensions of morphism Lie algebras are characterized via cohomology.

A relationship between morphism Lie group and algebra cohomology is established.

Abstract

A morphism Lie algebra is a triple $(\mathfrak{g}, \mathfrak{h}, \phi)$ consisting of two Lie algebras $\mathfrak{g}, \mathfrak{h}$ and a Lie algebra homomorphism $\phi : \mathfrak{g} \rightarrow \mathfrak{h}$. We define representations and cohomology of morphism Lie algebras. As applications of our cohomology, we study some aspects of deformations, abelian extensions of morphism Lie algebras and classify skeletal morphism sh Lie algebras. Finally, we consider the cohomology of morphism Lie groups and find a relation with the cohomology of morphism Lie algebras.