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The quantum Grothendieck ring of a certain category of finite-dimensionalmodules over a quantum loop algebra associated with a complexfinite-dimensional simple Lie algebra $\mathfrak{g}$ has a quantum clusteralgebra structure of skew-symmetric type. Partly motivated by a search of aring corresponding to a quantum cluster algebra of {\em skew-symmetrizable}type, the quantum {\em virtual} Grothendieck ring, denoted by$\mathfrak{K}_q(\mathfrak{g})$, is recently introduced by Kashiwara--Oh\cite{KO23} as a subring of the quantum torus based on the $(q,t)$-Cartanmatrix specialized at $q=1$. In this paper, we prove that$\mathfrak{K}_q(\mathfrak{g})$ indeed has a quantum cluster algebra structureof skew-symmetrizable type. This task essentially involves constructingdistinguished bases of $\mathfrak{K}_q(\mathfrak{g})$ that will be used to makecluster variables and generalizing the quantum $T$-system associated withKirillov--Reshetikhin modules to establish a quantum exchange relation ofcluster variables. Furthermore, these distinguished bases naturally fit intothe paradigm of Kazhdan--Lusztig theory and our study of these bases leads tosome conjectures on quantum positivity and $q$-commutativity.

Quantization of virtual Grothendieck rings and their structure including quantum cluster algebras