Representations of centrally extended Lie superalgebra $\mathfrak{psl}(2|2)$

TL;DR

This paper provides a comprehensive classification of finite-dimensional irreducible representations of the centrally extended Lie superalgebra (2|2), including new modules with degenerate eigenvalues, and constructs explicit bases using Mickelsson--Zhelobenko algebra techniques.

Contribution

It extends previous work by fully describing all finite-dimensional irreducible representations of (2|2), introducing new modules and explicit basis constructions.

Findings

Complete classification of irreducible representations.

Introduction of modules with degenerate central eigenvalues.

Explicit basis construction for all irreducible modules.

Abstract

The symmetries provided by representations of the centrally extended Lie superalgebra $\mathfrak{psl}(2|2)$ are known to play an important role in the spin chain models originated in the planar anti-de Sitter/conformal field theory correspondence and one-dimensional Hubbard model. We give a complete description of finite-dimensional irreducible representations of this superalgebra thus extending the work of Beisert which deals with a generic family of representations. Our description includes a new class of modules with degenerate eigenvalues of the central elements. Moreover, we construct explicit bases in all irreducible representations by applying the techniques of Mickelsson--Zhelobenko algebras.