Commutative 2-cocycles on Lie algebras

TL;DR

This paper investigates commutative 2-cocycles on Lie algebras, exploring their properties, relationships with antiderivations, and explicit computations for various classes including semisimple, current, and Kac-Moody algebras.

Contribution

It introduces the study of symmetric bilinear forms satisfying cocycle conditions on Lie algebras and computes these forms for several important classes.

Findings

Characterization of commutative 2-cocycles on Lie algebras

Relationship between commutative 2-cocycles and antiderivations

Explicit computations for semisimple, current, and Kac-Moody Lie algebras

Abstract

On Lie algebras, we study commutative 2-cocycles, i.e., symmetric bilinear forms satisfying the usual cocycle equation. We note their relationship with antiderivations and compute them for some classes of Lie algebras, including finite-dimensional semisimple, current and Kac-Moody algebras.