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

Abstract

In this paper, we investigate finite-dimensional irreducible representations of the quantum affine general linear superalgebra ${\mathrm{U}}_q\left({\widehat{\mathfrak{gl}}}_{m|n,\mathbf{s}}\right)$ for arbitrary 01-sequences $\mathbf{s}$, using the RTT presentation. We systematically construct the RTT presentation for quantum general linear superalgebra ${\mathrm{U}}_q\left({\mathfrak{gl}}_{m|n,\mathbf{s}}\right)$, 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 ${\mathrm{U}}_q\left({\mathfrak{gl}}_{m|n,\mathbf{s}}\right)$ 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.

arXiv comment42 pages. Comments are welcome!