Super duality and Crystal bases for quantum orthosymplectic superalgebras

TL;DR

This paper develops a super duality framework for quantum orthosymplectic superalgebras, classifies their irreducible modules, and introduces ortho-symplectic tableaux to describe crystal bases and graphs.

Contribution

It introduces a new tensor category for modules over quantum orthosymplectic superalgebras, classifies irreducible modules, and provides explicit combinatorial models for crystal bases.

Findings

Classification of irreducible modules in the category.

Existence of unique crystal bases for certain types.

Introduction of ortho-symplectic tableaux for crystal graph description.

Abstract

We introduce a semisimple tensor category $\mc{O}^{int}_q(m|n)$ of modules over an quantum ortho-symplectic superalgebra. It is a natural counterpart of the category of finitely dominated integrable modules over the quantum classical (super) algebra of type $B_{m+n}$, $C_{m+n}$, $D_{m+n}$ or $B(0,m+n)$ from a viewpoint of super duality. We classify the irreducible modules in $\mc{O}^{int}_q(m|n)$ and show that an irreducible module in $\mc{O}^{int}_q(m|n)$ has a unique crystal base in case of type $B$ and $C$. An explicit description of the crystal graph is given in terms of a new combinatorial object called ortho-symplectic tableaux.