This is a report on a joint work with Christof Geiss and David Hernandez (arXiv:2401.04616). Shifted quantum affine algebras have been introduced by Finkelberg and Tsymbaliuk in their study of quantized K-theoretic Coulomb branches of certain quiver gauge theories. Their representation theory has been studied by Hernandez, who constructed a category O containing finite-dimensional and infinite-dimensional representations. We introduce a new class of infinite rank cluster algebras associated with A-D-E root systems, and show that suitable completions of these cluster algebras are isomorphic to the Grothendieck rings of the category O of the corresponding shifted quantum affine algebras. In this isomorphism, the cluster variables of the initial seed are mapped to certain Q-variables and the most interesting first step mutations are instances of the QQ-system relations studied recently by Frenkel and Hernandez (arXiv:2312.13256). We conjecture that the images of all cluster monomials are classes of simple objects of O. We prove the conjecture in type A_1. We also show that it holds for the subcategory C of O whose objects are the finite-dimensional representations.

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