Quantum Queer Superalgebra and Crystal Bases

TL;DR

This paper develops the crystal basis theory for the quantum queer superalgebra, defining crystal bases, tensor product rules, and establishing their existence and uniqueness for finite-dimensional modules.

Contribution

It introduces the notion of crystal bases for the quantum queer superalgebra and proves their existence and uniqueness in a specific module category.

Findings

Defined crystal bases for $ ext{U}_q$

Established tensor product rule for crystal bases

Proved existence and uniqueness of crystal bases

Abstract

In this paper, we develop the crystal basis theory for the quantum queer superalgebra $\Uq$. We define the notion of crystal bases, describe the tensor product rule, and present the existence and uniqueness of crystal bases for finite-dimensional $\Uq$-modules in the category $\mathcal{O}_{int}^{\ge 0}$.