Ground state representations of loop algebras

TL;DR

This paper classifies translation-invariant 2-cocycles on a subalgebra of loop algebras and shows the uniqueness of ground state representations, linking them to vacuum representations of the loop algebra.

Contribution

It provides a classification of 2-cocycles on a subalgebra of loop algebras and establishes the uniqueness of ground state representations corresponding to vacuum states.

Findings

Classification of translation-invariant 2-cocycles on Sg

Uniqueness of ground state representations for each cocycle

Ground states correspond to vacuum representations of Lg

Abstract

Let g be a simple Lie algebra, Lg be the loop algebra of g. Fixing a point in S^1 and identifying the real line with the punctured circle, we consider the subalgebra Sg of Lg of rapidly decreasing elements on R. We classify the translation-invariant 2-cocycles on Sg. We show that the ground state representation of Sg is unique for each cocycle. These ground states correspond precisely to the vacuum representations of Lg.