Invariants of Lie algebras extended over commutative algebras without unit

TL;DR

This paper investigates the invariants of Lie algebras extended over commutative algebras without units, providing a unified approach to cohomology, bilinear forms, and derivations relevant to various algebraic structures.

Contribution

It introduces a unified framework for analyzing cohomology, invariant forms, and derivations of such Lie algebra extensions, connecting several previously separate topics.

Findings

Results on second cohomology with trivial coefficients

Characterization of symmetric invariant bilinear forms

Descriptions of derivations of extended Lie algebras

Abstract

We establish results about the second cohomology with coefficients in the trivial module, symmetric invariant bilinear forms and derivations of a Lie algebra extended over a commutative associative algebra without unit. These results provide a simple unified approach to a number of questions treated earlier in completely separated ways: periodization of semisimple Lie algebras (Anna Larsson), derivation algebras, with prescribed semisimple part, of nilpotent Lie algebras (Benoist), and presentations of affine Kac-Moody algebras.