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Monoidal categorification of cluster algebras

Authors: Seok-Jin Kang, Masaki Kashiwara, Myungho Kim and Se-jin Oh
Journal: J. Amer. Math. Soc. 31 (2018), 349-426
MSC (2010): Primary 13F60, 81R50, 16Gxx, 17B37
Published electronically: December 5, 2017
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Abstract: 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.

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Additional Information

Seok-Jin Kang
Affiliation: Research Institute of Computers, Information and Communication, Pusan National University, 2, Busandaehak-ro Pusan 46241, Korea

Masaki Kashiwara
Affiliation: Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan

Myungho Kim
Affiliation: Department of Mathematics, Kyung Hee University, Seoul 02447, Korea

Se-jin Oh
Affiliation: Department of Mathematics Ewha Womans University, Seoul 03760, Korea

Keywords: Cluster algebra, quantum cluster algebra, monoidal categorification, Khovanov--Lauda--Rouquier algebra, unipotent quantum coordinate ring, quantum affine algebra
Received by editor(s): February 15, 2015
Received by editor(s) in revised form: December 19, 2016, and July 15, 2017
Published electronically: December 5, 2017
Additional Notes: 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.
Article copyright: © Copyright 2017 American Mathematical Society

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