Cluster algebras and category O for representations of Borel subalgebras of quantum affine algebras

Abstract

Let $\mathcal{O}$ be the category of representations of the Borel subalgebra of a quantum affine algebra introduced by Jimbo and the first author. We show that the Grothendieck ring of a certain monoidal subcategory of $\mathcal{O}$ has the structure of a cluster algebra of infinite rank, with an initial seed consisting of prefundamental representations. In particular, the celebrated Baxter relations for the 6-vertex model get interpreted as Fomin-Zelevinsky mutation relations.

35 pages. v2 : Section 7.4 added

References in corpus (3)

• Hidden Grassmann Structure in the XXZ Model II: Creation Operators
• On minimal affinizations of representations of quantum groups
• The Baxter's Q-operator for the W-algebra $W_N$

Cited by in corpus (14)

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• A Q-operator for open spin chains I: Baxter's TQ relation
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• Quantum Grothendieck rings as quantum cluster algebras
• Length-two representations of quantum affine superalgebras and Baxter operators
• Yangians and Baxter's relations
• Dominance order and monoidal categorification of cluster algebras
• Unipotent quantum coordinate ring and prefundamental representations for types $A_n^{(1)}$ and $D_n^{(1)}$
• Elliptic quantum groups and Baxter relations
• A functor for constructing $R$-matrices in the category $\mathcal{O}$ of Borel quantum loop algebras
• Cluster realization of Weyl groups and $q$-characters of quantum affine algebras
• q-Opers, QQ-systems, and Bethe Ansatz II: Generalized Minors
• Inflations for representations of shifted quantum affine algebras