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

TL;DR

This paper proves that the quantum virtual Grothendieck ring has a skew-symmetrizable quantum cluster algebra structure, constructing bases and relations that extend known quantum systems and suggest positivity and commutativity conjectures.

Contribution

The paper establishes the quantum cluster algebra structure of the quantum virtual Grothendieck ring and constructs distinguished bases to facilitate this, extending quantum $T$-systems.

Findings

Quantum virtual Grothendieck ring has a skew-symmetrizable quantum cluster algebra structure.

Constructed distinguished bases fitting Kazhdan--Lusztig theory.

Formulated conjectures on quantum positivity and $q$-commutativity.

Abstract

The quantum Grothendieck ring of a certain category of finite-dimensional modules over a quantum loop algebra associated with a complex finite-dimensional simple Lie algebra $\mathfrak{g}$ has a quantum cluster algebra structure of skew-symmetric type. Partly motivated by a search of a ring 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 $(q,t)$-Cartan matrix specialized at $q=1$. In this paper, we prove that $\mathfrak{K}_q(\mathfrak{g})$ indeed has a quantum cluster algebra structure of skew-symmetrizable type. This task essentially involves constructing distinguished bases of $\mathfrak{K}_q(\mathfrak{g})$ that will be used to make cluster variables and generalizing the quantum $T$-system associated with Kirillov--Reshetikhin modules to establish a quantum exchange relation of cluster variables. Furthermore, these distinguished bases naturally fit into the paradigm of Kazhdan--Lusztig theory and our study of these bases leads to some conjectures on quantum positivity and $q$-commutativity.