Examples of quantum cluster algebras associated to partial flag varieties

TL;DR

This paper provides explicit examples of quantum cluster algebra structures on quantized coordinate rings of partial flag varieties and their unipotent radicals, extending classical cluster algebra structures to the quantum setting.

Contribution

It introduces new quantum cluster algebra structures on quantized coordinate rings and enveloping algebras, generalizing previous classical results.

Findings

Quantum cluster structures on partial flag varieties' coordinate rings

Quantum cluster structures on quantized enveloping algebras

Quantizations of classical cluster algebra structures

Abstract

We give several explicit examples of quantum cluster algebra structures, as introduced by Berenstein and Zelevinsky, on quantized coordinate rings of partial flag varieties and their associated unipotent radicals. These structures are shown to be quantizations of the cluster algebra structures found on the corresponding classical objects by Geiss, Leclerc and Schroer, whose work generalizes that of several other authors. We also exhibit quantum cluster algebra structures on the quantized enveloping algebras of the Lie algebras of the unipotent radicals.