Monoidal categorification and quantum affine algebras

2019/10/31 by Masaki Kashiwara, Myungho Kim, Se-jin Oh +2 · 37 citations
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Affine transformation #Algebra over a field #Algebra representation #Algebraic structures and combinatorial models #Categorification #Cellular algebra #Closed monoidal category #Cluster algebra #Discrete mathematics #Enriched category #Functor #Mathematics #Monoidal category #Nonlinear Waves and Solitons #Pure mathematics #Quantum #Quantum affine algebra #Symmetric monoidal category #math.QA #math.RT #msc:13F60 #msc:17B37 #msc:18D10

paper · pdf · doi:10.1112/s0010437x20007137

published in Compositio Mathematica 156(5), 1039-1077 (Cambridge University Press) · 42 pages

openalex publication_date 2020/04/27 · arxiv created 2020/09/29 · arxiv updated 2020/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We introduce and investigate new invariants of pairs of modules M and N over quantum affine algebras U_q^′ (\mathfrakg) by analyzing their associated R -matrices. Using these new invariants, we provide a criterion for a monoidal category of finite-dimensional integrable U_q^′ (\mathfrakg) -modules to become a monoidal categorification of a cluster algebra.

Citations

• Quiver varieties and finite dimensional representations of quantum affine algebras
• 2-Kac-Moody algebras
• Cluster structures on quantum coordinate rings
• A diagrammatic approach to categorification of quantum groups II
• Simple tensor products
• Existence of Kirillov–Reshetikhin crystals for nonexceptional types
• Simplicity of heads and socles of tensor products
• The Berenstein-Zelevinsky quantum cluster algebra conjecture
• Braid Group Actions and Tensor Products
• Simplicity of tensor products of Kirillov--Reshetikhin modules: nonexceptional affine and G types
• Categories over quantum affine algebras and monoidal categorification
• Canonical Bases Arising from Quantized Enveloping Algebras

Cited by

• Q-data and Representation Theory of Untwisted Quantum Affine Algebras