Monoidal categorification and quantum affine algebras II

• 成果类型: Article
• 署名作者: Kashiwara, Masaki; Kim, Myungho; Oh, Se-jin; Park, Euiyong
• 署名单位: Kyoto University; Korea Institute for Advanced Study (KIAS); Kyung Hee University; Sungkyunkwan University (SKKU); University of Seoul
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-024-01249-1
• 发表日期: 2024
• 页码: 837-924
• 关键词: QUIVER HECKE ALGEBRAS; CLUSTER ALGEBRAS; R-MATRICES; Q-CHARACTERS; T-SYSTEMS; REPRESENTATION; MODULES; PERIODICITIES; CATEGORIES; VARIETIES

摘要

We introduce a new family of real simple modules over the quantum affine algebras, called the affine determinantial modules, which contains the Kirillov-Reshetikhin (KR)-modules as a special subfamily, and then prove T-systems among them which generalize the T-systems among KR-modules and unipotent quantum minors in the quantum unipotent coordinate algebras simultaneously. We develop new combinatorial tools: admissible chains of i\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$i$\end{document}-boxes which produce commuting families of affine determinantial modules, and box moves which describe the T-system in a combinatorial way. Using these results, we prove that various module categories over the quantum affine algebras provide monoidal categorifications of cluster algebras. As special cases, Hernandez-Leclerc categories Cg0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathscr{C}_{{\mathfrak{g}}}<^>{0}$\end{document} and Cg-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathscr{C}_{{\mathfrak{g}}}<^>{-}$\end{document} provide monoidal categorifications of the cluster algebras for an arbitrary quantum affine algebra.