Representations of shifted quantum affine algebras and cluster algebras I. The simply-laced case

Abstract

We introduce a family of cluster algebras of infinite rank associated with root systems of type $A$, $D$, $E$. We show that suitable completions of these cluster algebras are isomorphic to the Grothendieck rings of the categories ${\mathcal{O}}_{\mathbb{Z}}$ of the corresponding shifted quantum affine algebras. The cluster variables of a class of distinguished initial seeds are certain formal power series defined by E. Frenkel and the second author, which satisfy a system of functional relations called $QQ$-system. We conjecture that all cluster monomials are classes of simple objects of ${\mathcal{O}}_{\mathbb{Z}}$. In the final section, we show that these cluster algebras contain infinitely many cluster subalgebras isomorphic to the coordinate ring of the open double Bruhat cell of the corresponding simple simply-connected algebraic group. This explains the similarity between $QQ$-system relations and certain generalized minor identities discovered by Fomin and Zelevinsky.

68 pages, 20 figures. V2 : references added, improved exposition in Sections 10.3, 10.4. V3 : 70 pages. Minor corrections and improvements. Final version to appear in PLMS