Monoidal categorification of cluster algebras

Seok Jin Kang, Masaki Kashiwara, Myungho Kim, Se Jin Oh

Research output: Contribution to journalArticlepeer-review

45 Scopus citations

Abstract

We prove that the quantum cluster algebra structure of a unipotent quantum coordinate ring Aq(n(w)), associated with a symmetric Kac-Moody algebra and its Weyl group element w, admits a monoidal categorification via the representations of symmetric Khovanov-Lauda-Rouquier algebras. In order to achieve this goal, we give a formulation of monoidal categorifications of quantum cluster algebras and provide a criterion for a monoidal category of finite-dimensional graded R-modules to become a monoidal categorification, where R is a symmetric Khovanov-Lauda-Rouquier algebra. Roughly speaking, this criterion asserts that a quantum monoidal seed can be mutated successively in all the directions, once the first-step mutations are possible. Then, we show the existence of a quantum monoidal seed of Aq(n(w)) which admits the first-step mutations in all the directions. As a consequence, we prove the conjecture that any cluster monomial is a member of the upper global basis up to a power of q1/2. In the course of our investigation, we also give a proof of a conjecture of Leclerc on the product of upper global basis elements.

Original languageEnglish
Pages (from-to)349-426
Number of pages78
JournalJournal of the American Mathematical Society
Volume31
Issue number2
DOIs
StatePublished - Apr 2018

Keywords

  • Cluster algebra
  • Khovanov-Lauda-Rouquier algebra
  • Monoidal categorification
  • Quantum affine algebra
  • Quantum cluster algebra
  • Unipotent quantum coordinate ring

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