Quantization of Virtual Grothendieck Rings and Their Structure Including Quantum Cluster Algebras

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 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 skew-symmetrizable type, the quantum virtual Grothendieck ring, denoted by K_q(g), is recently introduced by Kashiwara and Oh (Math Z 303(2):42, 2023) as a subring of the quantum torus based on the (q, t)-Cartan matrix specialized at q=1. In this paper, we prove that K_q(g) indeed has a quantum cluster algebra structure of skew-symmetrizable type. This task essentially involves constructing distinguished bases of K_q(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.