Monoidal categorification of cluster algebras

Kang, Seok-Jin¹ ; Kashiwara, Masaki² ; Kim, Myungho³ ; Oh, Se-jin⁴

¹ Research Institute of Computers, Information and Communication, Pusan National University, 2, Busandaehak-ro Pusan 46241, Korea
² Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan
³ Department of Mathematics, Kyung Hee University, Seoul 02447, Korea
⁴ Department of Mathematics Ewha Womans University, Seoul 03760, Korea

Journal of the American Mathematical Society, Tome 31 (2018) no. 2, pp. 349-426

Résumé

We prove that the quantum cluster algebra structure of a unipotent quantum coordinate ring $A_q(\mathfrak {n}(w))$, 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 $A_q(\mathfrak {n}(w))$ 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 $q^{1/2}$. In the course of our investigation, we also give a proof of a conjecture of Leclerc on the product of upper global basis elements.

Détail

DOI : 10.1090/jams/895

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