Cluster algebras and category \(\mathcal O\) for representations of Borel subalgebras of quantum affine algebras. (English) Zbl 1390.17025
Summary: 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.
MSC:
17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
13F60 | Cluster algebras |
17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
82B23 | Exactly solvable models; Bethe ansatz |