Categorical crystals for quantum affine algebras

Masaki Kashiwara; Euiyong Park

doi:10.1515/crelle-2022-0061

Categorical crystals for quantum affine algebras

Masaki Kashiwara, Euiyong Park

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

In this paper, a new categorical crystal structure for the quantum affine algebras is presented. We introduce the notion of extended crystals B_g(∞) for an arbitrary quantum group U_q(g), which is the product of infinite copies of the crystal B(∞). For a complete duality datum D in the Hernandez–Leclerc category C_g⁰ of a quantum affine algebra U_q^′ (g), we prove that the set B_D(g) of the isomorphism classes of simple modules in C_g⁰ has an extended crystal structure isomorphic to B^b_gfin(∞). An explicit combinatorial description of the extended crystal B_D(g) for affine type A⁽¹⁾_n is given in terms of affine highest weights.

Original language	English
Pages (from-to)	223-267
Number of pages	45
Journal	Journal fur die Reine und Angewandte Mathematik
Volume	2022
Issue number	792
DOIs	https://doi.org/10.1515/crelle-2022-0061
State	Published - 1 Nov 2022

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N2 - In this paper, a new categorical crystal structure for the quantum affine algebras is presented. We introduce the notion of extended crystals Bg(∞) for an arbitrary quantum group Uq(g), which is the product of infinite copies of the crystal B(∞). For a complete duality datum D in the Hernandez–Leclerc category Cg0 of a quantum affine algebra Uq′ (g), we prove that the set BD(g) of the isomorphism classes of simple modules in Cg0 has an extended crystal structure isomorphic to Bbgfin(∞). An explicit combinatorial description of the extended crystal BD(g) for affine type A(1)n is given in terms of affine highest weights.

AB - In this paper, a new categorical crystal structure for the quantum affine algebras is presented. We introduce the notion of extended crystals Bg(∞) for an arbitrary quantum group Uq(g), which is the product of infinite copies of the crystal B(∞). For a complete duality datum D in the Hernandez–Leclerc category Cg0 of a quantum affine algebra Uq′ (g), we prove that the set BD(g) of the isomorphism classes of simple modules in Cg0 has an extended crystal structure isomorphic to Bbgfin(∞). An explicit combinatorial description of the extended crystal BD(g) for affine type A(1)n is given in terms of affine highest weights.

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Categorical crystals for quantum affine algebras

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