Formal deformations and extensions of `twisted' Lie algebras

TL;DR

This paper studies the formal deformation and extension theory of twisted Lie algebras influenced by derivations, introducing new cohomology frameworks and analyzing their algebraic and cohomological properties.

Contribution

It defines InvDer Lie, develops its cohomology theory, and explores central extensions, advancing understanding of deformations of twisted Lie algebras.

Findings

Defined representations of InvDer Lie

Established cohomology structures for InvDer Lie

Linked central extensions to cohomology theory

Abstract

The interplay between derivations and algebraic structures has been a subject of significant interest and exploration. Inspired by Yau's twist and the Leibniz rule, we investigate the formal deformation of twisted Lie algebras by invertible derivations, herein referred to as "InvDer Lie". We define representations of InvDer Lie, elucidate cohomology structures of order 1 and 2, and identify infinitesimals as 2-cocycles. Furthermore, we explore central extensions of InvDer Lie, revealing their intricate relationship with cohomology theory.