Cluster structures on quantum coordinate rings

Understand

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.