Cluster structures on quantum coordinate rings

TL;DR

This paper demonstrates that quantum coordinate rings of certain subgroups in Kac-Moody groups possess a quantum cluster algebra structure, derived from module categories and quantum determinantal identities, extending to partial flag varieties.

Contribution

It establishes a quantum cluster algebra structure on quantum coordinate rings of unipotent subgroups and partial flag varieties, linking representation theory and quantum algebra.

Findings

Quantum coordinate rings have a natural quantum cluster structure.

Quantum determinantal identities serve as q-analogues of T-systems.

Results apply to simple algebraic groups of types A, D, E.

Abstract

We show that the quantum coordinate ring of the unipotent subgroup N(w) of a symmetric Kac-Moody group G associated with a Weyl group element w has the structure of a quantum cluster algebra. This quantum cluster structure arises naturally from a subcategory C_w of the module category of the corresponding preprojective algebra. An important ingredient of the proof is a system of quantum determinantal identities which can be viewed as a q-analogue of a T-system. In case G is a simple algebraic group of type A, D, E, we deduce from these results that the quantum coordinate ring of an open cell of a partial flag variety attached to G also has a cluster structure.