Monoidal categorification of cluster algebras

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

Research output: Contribution to journalArticlepeer-review

60 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

Bibliographical note

Funding Information:
This work was supported by Grant-in-Aid for Scientific Research (B) 22340005, Japan Society for the Promotion of Science. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. NRF-2017R1C1B2007824). This work was supported by NRF Grant # 2016R1C1B2013135. This research was supported by Ministry of Culture, Sports and Tourism (MCST) and Korea Creative Content Agency (KOCCA) in the Culture Technology (CT) Research & Development Program 2017.

Funding Information:
Received by the editors February 15, 2015, and in revised form, December 19, 2016, and July 15, 2017. 2010 Mathematics Subject Classification. Primary 13F60, 81R50, 16Gxx, 17B37. Key words and phrases. Cluster algebra, quantum cluster algebra, monoidal categorification, Khovanov–Lauda–Rouquier algebra, unipotent quantum coordinate ring, quantum affine algebra. This work was supported by Grant-in-Aid for Scientific Research (B) 22340005, Japan Society for the Promotion of Science. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. NRF-2017R1C1B2007824). This work was supported by NRF Grant # 2016R1C1B2013135. This research was supported by Ministry of Culture, Sports and Tourism (MCST) and Korea Creative Content Agency (KOCCA) in the Culture Technology (CT) Research & Development Program 2017.

Publisher Copyright:
© 2017 American Mathematical Society.

Keywords

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

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