Monoidal categorification of cluster algebras

Kang, Seok Jin; Kashiwara, Masaki; Kim, Myung Ho; Oh, Se Jin

doi:10.1090/jams/895

Cited by 104 publications

(206 citation statements)

References 43 publications

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“…Kang, Kashiwara, Kim, and Oh [36] proved that the quantum unipotent coordinate algebra has a monoidal categorification as conjectured in [26,38]. The connection between monoidal categorification and quantum affine algebras is as follows.…”

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confidence: 87%

Quantum affine algebras and Grassmannians

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We study the relation between quantum affine algebras of type A and Grassmannian cluster algebras. Hernandez and Leclerc described an isomorphism from the Grothendieck ring of a certain subcategory C ℓ of U q ( sl n )-modules to a quotient of the Grassmannian cluster algebra in which certain frozen variables are set to 1. We explain how this induces an isomorphism between the monoid of dominant monomials, used to parameterize simple modules, and a quotient of the monoid of rectangular semistandard Young tableaux with n rows and with entries in [n + ℓ + 1]. Via the isomorphism, we define an element ch(T ) in a Grassmannian cluster algebra for every rectangular tableau T . By results of Kashiwara, Kim, Oh, and Park, and also of Qin, every Grassmannian cluster monomial is of the form ch(T ) for some T . Using a formula of Arakawa-Suzuki, we give an explicit expression for ch(T ), and also give explicit q-character formulas for finite-dimensional U q ( sl n )-modules. We give a tableau-theoretic rule for performing mutations in Grassmannian cluster algebras. We suggest how our formulas might be used to study reality and primeness of modules, and compatibility of cluster variables.

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Quantum affine algebras and Grassmannians

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“…The following lemma can be proved similarly to the quiver Hecke algebra case [, Proposition 3.2.9], and we do not repeat the proof. Lemma Let

V, W

be simple modules in

C_{g}

and assume that one of them is real.…”

Section: Symmetric Quiver Hecke Algebras and Quantum Affine Algebrasmentioning

confidence: 99%

Monoidal categories of modules over quantum affine algebras of type A and B

Kashiwara

2018

Proc. London Math. Soc.

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We construct an exact tensor functor from the category scriptA of finite‐dimensional graded modules over the quiver Hecke algebra of type A∞ to the category CBn(1) of finite‐dimensional integrable modules over the quantum affine algebra of type Bnfalse(1false). It factors through the category T2n, which is a localization of scriptA. As a result, this functor induces a ring isomorphism from the Grothendieck ring of T2n (ignoring the gradings) to the Grothendieck ring of a subcategory CBn(1)0 of CBn(1). Moreover, it induces a bijection between the classes of simple objects. Because the category T2n is related to categories CA2n−1(t)0 (t=1,2) of the quantum affine algebras of type A2n−1false(tfalse), we obtain an interesting connection between those categories of modules over quantum affine algebras of type A and type B. Namely, for each t=1,2, there exists an isomorphism between the Grothendieck ring of CA2n−1(t)0 and the Grothendieck ring of CBn(1)0, which induces a bijection between the classes of simple modules.

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“…Proof If