Inonu-Wigner Contractions of Kac-Moody Algebras

TL;DR

This paper explores the Inönü-Wigner contractions of affine Kac-Moody algebras, revealing conditions for the Sugawara construction and deriving a Virasoro algebra from coset space contractions.

Contribution

It provides a detailed analysis of contractions of affine Kac-Moody algebras and their impact on the Sugawara construction and Virasoro algebra derivation.

Findings

Sugawara construction exists only at a fixed level k after contraction

Contractions of G/H coset spaces lead to an affine translation algebra

Derived Virasoro algebra has central charge equal to dim(G/H)

Abstract

We discuss In\"on\"u-Wigner contractions of affine Kac-Moody algebras. We show that the Sugawara construction for the contracted affine algebra exists only for a fixed value of the level $k$, which is determined in terms of the dimension of the uncontracted part of the starting Lie algebra, and the quadratic Casimir in the adjoint representation. Further, we discuss contractions of $G/H$ coset spaces, and obtain an affine {\it translation} algebra, which yields a Virasoro algebra (via a GKO construction) with a central charge given by $dim(G/H)$.