Categorical crystals for quantum affine algebras

Abstract

A new categorical crystal structure for the quantum affine algebras is presented. We introduce the extended crystal ${\hat{B}}_{\mathfrak{g}}(\infty )$ 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 ${\mathcal{C}}_{\mathfrak{g}}^0$ of a quantum affine algebra $U_q^{\prime }(\mathfrak{g})$, we prove that the set of the isomorphism classes of simple modules in ${\mathcal{C}}_{\mathfrak{g}}^0$ has an extended crystal structure isomorphic to the extended crystal ${\hat{B}}_{\mathfrak{g}}(\infty )$. An explicit combinatorial description of the extended crystal ${\hat{B}}_{\mathfrak{g}}(\infty )$ for affine type $A_n^{(1)}$ is given in terms of affine highest weights.

arXiv comment61 pages

References in corpus (7)

• A diagrammatic approach to categorification of quantum groups I
• 2-Kac-Moody algebras
• Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras
• Categorification of Highest Weight Modules via Khovanov-Lauda-Rouquier Algebras
• Crystals from categorified quantum groups
• Monoidal categorification and quantum affine algebras
• Twisted and folded Auslander-Reiten quivers and applications to the representation theory of quantum affine algebras