Representations of Quantum Affine General Linear Superalgebras at Arbitrary 01-Sequences

TL;DR

This paper classifies finite-dimensional irreducible representations of quantum affine superalgebras for arbitrary 01-sequences, extending previous frameworks and providing explicit conditions and constructions.

Contribution

It introduces a systematic construction of the RTT presentation and classifies finite-dimensional irreducible representations for arbitrary 01-sequences in quantum affine superalgebras.

Findings

Derived PBW basis compatible with non-standard parities

Established necessary and sufficient conditions for finite-dimensionality

All finite-dimensional representations for (m,n)=(1,1) are tensor products of evaluation representations

Abstract

In this paper, we investigate finite-dimensional irreducible representations of the quantum affine general linear superalgebra $\mathrm{U}_q\big(\widehat{\mathfrak{gl}}_{m|n,\mathbf{s}}\big)$ for arbitrary 01-sequences $\mathbf{s}$, using the RTT presentation. We systematically construct the RTT presentation for quantum general linear superalgebra $\mathrm{U}_q\big(\mathfrak{gl}_{m|n,\mathbf{s}}\big)$, and derive a PBW basis induced by the action of the braid group, compatible with non-standard parities. We determine the necessary and sufficient conditions for the finite-dimensionality of irreducible representations of $\mathrm{U}_q\big(\mathfrak{gl}_{m|n,\mathbf{s}}\big)$ and extend the results to the affine case via the evaluation homomorphism. Specific cases such as $(m,n)=(1,1)$ demonstrate that all finite-dimensional representations are tensor products of typical evaluation representations. This work extends existing representation frameworks and classification methods to encompass arbitrary 01-sequences, establishing the foundation for subsequent research on representations of quantum affine superalgebras.