Categorical crystals for quantum affine algebras

TL;DR

This paper introduces a new categorical crystal structure for quantum affine algebras, extending the crystal $B() over infinite copies and providing explicit combinatorial descriptions for affine type $A_n^{(1)}$.

Contribution

It constructs the extended crystal $\u00afB_{\u0000g}(f)$ for quantum groups and proves its isomorphism with simple modules in Hernandez-Leclerc categories.

Findings

Defined the extended crystal structure for quantum affine algebras.

Proved the isomorphism with simple modules in the Hernandez-Leclerc category.

Provided explicit combinatorial descriptions for affine type $A_n^{(1)}$.

Abstract

A new categorical crystal structure for the quantum affine algebras is presented. We introduce the extended crystal $\widehat{B}_{\mathfrak{g}}(\infty)$ for an arbitrary quantum group, which is the product of infinite copies of the crystal $B(\infty)$. For a complete duality datum in the Hernandez-Leclerc category $\mathcal{C}^0_{\mathfrak{g}}$ of a quantum affine algebra $U_q'(\mathfrak{g})$, we prove that the set of the isomorphism classes of simple modules in $\mathcal{C}^0_{\mathfrak{g}}$ has an extended crystal structure isomorphic to the extended crystal $\widehat{B}_{\mathfrak{g}}(\infty)$. An explicit combinatorial description of the extended crystal $\widehat{B}_{\mathfrak{g}}(\infty)$ for affine type $A_n^{(1)}$ is given in terms of affine highest weights.