Integrable representations of the quantum affine special linear superalgebra

TL;DR

This paper classifies all simple integrable modules with finite-dimensional weight spaces for the quantum affine special linear superalgebra at generic q, revealing their highest or lowest weight structure depending on the level.

Contribution

It provides a complete classification of integrable modules for the quantum affine superalgebra, identifying conditions for their existence based on the parameters M and N.

Findings

Integrable modules are highest or lowest weight modules depending on the level.

Nonzero level integrable modules exist only if M or N equals 1.

Classification applies to modules with finite-dimensional weight spaces.

Abstract

The simple integrable modules with finite dimensional weight spaces are classified for the quantum affine special linear superalgebra $\U_q(\hat{\mathfrak{sl}}(M|N))$ at generic $q$. Any such module is shown to be a highest weight or lowest weight module with respect to one of the two natural triangular decompositions of the quantum affine superalgebra depending on whether the level of the module is zero or not. Furthermore, integrable $\U_q(\hat{\mathfrak{sl}}(M|N))$-modules at nonzero levels exist only if $M$ or $N$ is $1$.