The Extension Structure of 2D Massive Current Algebras

TL;DR

This paper investigates the extension structure of 2D massive current algebras in non-linear sigma models, revealing a two-step extension process involving abelian ideals and analyzing the associated cocycles.

Contribution

It introduces Kostant Sternberg $(L,M)$ systems to analyze the algebra's extension structure, identifying a non-split extension by abelian ideals and characterizing the cocycle involved.

Findings

The algebra exhibits a two-step extension by abelian ideals.

The second extension is a non-split extension of a representation.

The cocycle associated with the extension is thoroughly analyzed.

Abstract

The extension structure of the 2-dimensional current algebra of non-linear sigma models is analysed by introducing Kostant Sternberg $(L,M)$ systems. It is found that the algebra obeys a two step extension by abelian ideals. The second step is a non-split extension of a representation of the quotient of the algebra by the first step of the extension. The cocycle which appears is analysed.