Quantum cluster algebras and representations of shifted quantum affine algebras

Abstract We construct a new quantization $$K_t(\mathcal {O}^{\text {sh}}_{\mathbb {Z}})$$ K t ( O Z sh ) of the Grothendieck ring of the category $$\mathcal {O}^{\text {sh}}_{\mathbb {Z}}$$ O Z sh of representations of shifted quantum affine algebras (of simply-laced type). We establish that our quantization is compatible with the quantum Grothendieck ring $$K_t(\mathcal {O}^{\mathfrak {b},+}_{\mathbb {Z}})$$ K t ( O Z b , + ) for the quantum Borel affine algebra, namely that there is a natural embedding $$K_t(\mathcal {O}^{\mathfrak {b},+}_{\mathbb {Z}})\hookrightarrow K_t(\mathcal {O}^{\text {sh}}_{\mathbb {Z}})$$ K t ( O Z b , + ) ↪ K t ( O Z sh ) . Our construction is partially based on the cluster algebra structure on the classical Grothendieck ring discovered by Geiss–Hernandez–Leclerc. As first applications, we formulate a quantum analogue of QQ -systems (that we make completely explicit in type $$A_1$$ A 1 ). We also prove that the quantum oscillator algebra is isomorphic to a localization of a subalgebra of our quantum Grothendieck ring and that it is also isomorphic to the Berenstein–Zelevinsky’s quantum double Bruhat cell $$\mathbb {C}_t[{{\,\textrm{SL}\,}}_2^{w_0,w_0}]$$ C t [ SL 2 w 0 , w 0 ] .