Quantum cluster algebras and quantum nilpotent algebras

TL;DR

This paper proves that a broad class of noncommutative algebras and Poisson nilpotent algebras have canonical quantum cluster algebra structures, confirming conjectures and extending results in Lie theory and quantum groups.

Contribution

It establishes that many algebras in Lie theory possess canonical quantum cluster structures, confirming the Berenstein--Zelevinsky conjecture and extending quantum cluster algebra theory.

Findings

All algebras in the studied class have canonical quantum cluster structures.

Confirmed the Berenstein--Zelevinsky conjecture for quantized coordinate rings.

Constructed quantum cluster structures on quantum unipotent groups.

Abstract

A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large axiomatically defined class of noncommutative algebras possess canonical quantum cluster algebra structures. Furthermore, they coincide with the corresponding upper quantum cluster algebras. We also establish analogs of these results for a large class of Poisson nilpotent algebras. Many important families of coordinate rings are subsumed in the class we are covering, which leads to a broad range of application of the general results to the above mentioned types of problems. As a consequence, we prove the Berenstein--Zelevinsky conjecture for the quantized coordinate rings of double Bruhat cells and construct quantum cluster algebra structures on all quantum unipotent groups, extending the theorem of Gei\ss, Leclerc and Schr\"oer for the case of symmetric Kac--Moody groups. Moreover, we prove that the upper cluster algebras of Berenstein, Fomin and Zelevinsky associated to double Bruhat cells coincide with the corresponding cluster algebras.