Crystal base of the negative half of the quantum superalgebra $U_q(\mathfrak{gl}(m|n))$

TL;DR

This paper constructs and describes crystal bases for the negative half of the quantum superalgebra U_q(gl(m|n)), providing combinatorial models and compatibility results with related modules, advancing understanding of quantum superalgebra representations.

Contribution

It introduces explicit crystal bases for U_q(gl(m|n))^-, including combinatorial descriptions and compatibility with Kac modules and parabolic Verma modules.

Findings

Constructed crystal base of U_q(gl(m|n))^-

Provided combinatorial description of the crystal ty

Established compatibility with Kac and Verma modules

Abstract

We construct a crystal base of $U_q(\mathfrak{gl}(m|n))^-$, the negative half of the quantum superalgebra $U_q(\mathfrak{gl}(m|n))$. We give a combinatorial description of the associated crystal $\mathscr{B}_{m|n}(\infty)$, which is equal to the limit of the crystals of the ($q$-deformed) Kac modules $K(\lambda)$. We also construct a crystal base of a parabolic Verma module $X(\lambda)$ associated with the subalgebra $U_q(\mathfrak{gl}_{0|n})$, and show that it is compatible with the crystal base of $U_q(\mathfrak{gl}(m|n))^-$ and the Kac module $K(\lambda)$ under the canonical embedding and projection of $X(\lambda)$ to $U_q(\mathfrak{gl}(m|n))^-$ and $K(\lambda)$, respectively.