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

Hernandez, David; Leclerc, Bernard

doi:10.2140/ant.2016.10.2015

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Cluster algebras and category $\mathscr{O}$ for representations of Borel subalgebras of quantum affine algebras

David Hernandez and Bernard Leclerc

Vol. 10 (2016), No. 9, 2015–2052

DOI: 10.2140/ant.2016.10.2015

Abstract

Let $O$ 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 $O$ 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.

Keywords

quantum affine algebras, cluster algebras, category $\mathscr{O}$, monoidal categorification, Baxter relations\lower5pt\hbox

Mathematical Subject Classification 2010

Primary: 17B37

Secondary: 13F60, 17B10, 82B23

Milestones

Received: 19 April 2016

Accepted: 9 August 2016

Published: 22 November 2016

Authors

David Hernandez
Institut de Mathématiques de Jussieu-Paris Rive Gauche Université Paris-Diderot Paris 7 Bâtiment Sophie Germain, Case 7012 75205 Paris Cedex 13 France

Bernard Leclerc
Laboratoire de Mathématiques Nicolas Oresme Université de Caen Basse-Normandie 14032 Caen France

Vol. 10, No. 9, 2016