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

TL;DR

This paper constructs all typical finite-dimensional representations of the quantum superalgebra U_q[gl(2/2)] at generic q, extending classical representation theory to the quantum case and providing explicit module decompositions.

Contribution

It provides a complete construction of typical finite-dimensional representations of U_q[gl(2/2)] at generic q, including their decomposition into irreducible submodules.

Findings

All typical finite-dimensional representations are constructed explicitly.

Representations decompose into finite-dimensional irreducible U_q[gl(2)⊕gl(2)]-submodules.

The modules are irreducible and analogous to the classical case.

Abstract

In the present paper we construct all typical finite-dimensional representations of the quantum Lie superalgebra $U_{q}[gl(2/2)]$ at generic deformation parameter $q$. As in the non-deformed case the finite-dimensional $U_{q}[gl(2/2)]$-module $W^{q}$ obtained is irreducible and can be decomposed into finite-dimensional irreducible $U_{q}[gl(2)\oplus gl(2)]$-submodules $V^{q}_{k}$