Finite-Dimensional Representations of the Quantum Superalgebra U$_{q}$[gl(2/2)]: II. Nontypical representations at generic $q$

TL;DR

This paper constructs and classifies finite-dimensional nontypical representations of the quantum superalgebra U_q[gl(2/2)] at generic q, providing explicit matrix elements and analyzing their module structures.

Contribution

It extends previous work by explicitly constructing and classifying all nontypical finite-dimensional representations of U_q[gl(2/2)] at generic q, including explicit matrix elements.

Findings

Explicit matrix elements of all nontypical representations provided

Classification of indecomposable and nontypical modules achieved

Analysis of module structures and invariant submodules included

Abstract

The construction approach proposed in the previous paper Ref. 1 allows us there and in the present paper to construct at generic deformation parameter $q$ all finite--dimensional representations of the quantum Lie superalgebra $U_{q}[gl(2/2)]$. The finite--dimensional $U_{q}[gl(2/2)]$-modules $W^{q}$ constructed in Ref. 1 are either irreducible or indecomposible. If a module $W^{q}$ is indecomposible, i.e. when the condition (4.41) in Ref. 1 does not hold, there exists an invariant maximal submodule of $W^{q}$, to say $I_{k}^{q}$, such that the factor-representation in the factor-module $W^{q}/I_{k}^{q}$ is irreducible and called nontypical. Here, in this paper, indecomposible representations and nontypical finite--dimensional representations of the quantum Lie superalgebra $U_{q}[gl(2/2)]$ are considered and classified as their module structures are analized and the matrix elements of all nontypical representations are written down explicitly.