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

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 𝒪_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 𝒪_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.