Realizations of Affine Lie Algebra A_^(1) at Negative Levels

TL;DR

This paper presents a new realization of the affine Lie algebra A^(1)_1 at negative levels using parafermions, extending previous work on critical level realizations and enriching the algebraic understanding.

Contribution

It introduces a parafermion-based realization of affine Lie algebra A^(1)_1 at negative levels, expanding the algebraic framework beyond the critical level.

Findings

Realization of affine Lie algebra A^(1)_1 at negative levels using parafermions

Extension of previous critical level realization work

Provides new algebraic structures for negative level representations

Abstract

A realization of the affine Lie algebra ${A^{(1)}_1}$ and the relevant $Z$-algebra at negative level $-k$ is given in terms of parafermions. This generalizes the recent work on realization of the affine Lie algebra at the critical level.