The Berenstein-Zelevinsky quantum cluster algebra conjecture

TL;DR

This paper proves the Berenstein-Zelevinsky conjecture that the quantized coordinate rings of double Bruhat cells in simple algebraic groups have quantum cluster algebra structures, providing explicit seeds and establishing their properties.

Contribution

It confirms the conjecture for all finite dimensional simple algebraic groups, constructs explicit quantum seeds, and studies quantum double Bruhat cells and their properties.

Findings

Quantum coordinate rings admit quantum cluster algebra structures.

Upper quantum cluster algebras coincide with constructed quantum cluster algebras.

Explicit quantum seeds are provided for all double Bruhat cells.

Abstract

We prove the Berenstein-Zelevinsky conjecture that the quantized coordinate rings of the double Bruhat cells of all finite dimensional simple algebraic groups admit quantum cluster algebra structures with initial seeds as specified by [4]. We furthermore prove that the corresponding upper quantum cluster algebras coincide with the constructed quantum cluster algebras and exhibit a large number of explicit quantum seeds. Along the way a detailed study of the properties of quantum double Bruhat cells from the viewpoint of noncommutative UFDs is carried out and a quantum analog of the Fomin-Zelevinsky twist map is constructed and investigated for all double Bruhat cells. The results are valid over base fields of arbitrary characteristic and the deformation parameter is only assumed to be a non-root of unity.