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Monoidal categorification of cluster algebras
- Title
- Monoidal categorification of cluster algebras
- Authors
- Kang S.-J.; Kashiwara M.; Kim M.; Oh S.-J.
- Ewha Authors
- 오세진
- SCOPUS Author ID
- 오세진
- Issue Date
- 2018
- Journal Title
- Journal of the American Mathematical Society
- ISSN
- 0894-0347
- Citation
- Journal of the American Mathematical Society vol. 31, no. 2, pp. 349 - 426
- Keywords
- Cluster algebra; Khovanov-Lauda-Rouquier algebra; Monoidal categorification; Quantum affine algebra; Quantum cluster algebra; Unipotent quantum coordinate ring
- Publisher
- American Mathematical Society
- Indexed
- SCIE; SCOPUS
- Document Type
- Article
- 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. © 2017 American Mathematical Society.
- DOI
- 10.1090/JAMS/895
- Appears in Collections:
- 자연과학대학 > 수학전공 > Journal papers
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