Finite Dimensional Representations of the Quantum Superalgebra $U_q[gl(3/2)]$ in a Reduced $U_q[gl(3/2)] \supset U_q[gl(3/1)] \supset U_q[gl(3)]$ Basis

TL;DR

This paper provides explicit transformation formulas for finite-dimensional modules of the quantum superalgebra $U_q[gl(3/2)]$, using a Gel'fand-Zetlin-like basis, for generic deformation parameter $q$.

Contribution

It introduces explicit basis transformation formulas for all essentially typical finite-dimensional modules of $U_q[gl(3/2)]$, extending Gel'fand-Zetlin basis techniques to this quantum superalgebra.

Findings

Explicit transformation formulas for basis elements under Chevalley generators.

Basis resembles Gel'fand-Zetlin basis for $gl(5)$.

Results valid for generic $q$, covering typical modules.

Abstract

For generic $q$ we give expressions for the transformations of all essentially typical finite-dimensional modules of the Hopf superalgebra $U_q[gl(3/2)]$. The latter is a deformation of the universal enveloping algebra of the Lie superalgebra $gl(3/2)$. The basis within each module is similar to the Gel'fand-Zetlin basis for $gl(5)$. We write down expressions for the transformations of the basis under the action of the Chevalley generators.