Twisted cocycles of Lie algebras and corresponding invariant functions

TL;DR

This paper introduces a generalized framework for Lie algebra cocycles using complex parameters, leading to invariant functions that classify low-dimensional Lie algebras and analyze their contractions.

Contribution

It develops a new class of invariant functions based on parametric cocycles, enhancing classification and contraction analysis of Lie algebras.

Findings

Invariant functions classify 3- and 4-dimensional Lie algebras.

Functions help analyze 8-dimensional nilpotent continua.

Necessary criteria for 1-parametric contractions are established.

Abstract

We consider finite-dimensional complex Lie algebras. Using certain complex parameters we generalize the concept of cohomology cocycles of Lie algebras. A special case is generalization of 1-cocycles with respect to the adjoint representation - so called $(\alpha,\beta,\gamma)$--derivations. Parametric sets of spaces of cocycles allow us to define complex functions which are invariant under Lie isomorphisms. Such complex functions thus represent useful invariants - we show how they classify three and four-dimensional Lie algebras as well as how they apply to some eight-dimensional one-parametric nilpotent continua of Lie algebras. These functions also provide necessary criteria for existence of 1-parametric continuous contraction.