# Twist automorphisms on quantum unipotent cells and dual canonical bases

## TL;DR

This paper introduces quantum twist automorphisms on unipotent cells, demonstrating their preservation of dual canonical bases, their description via syzygy functors, and their compatibility with quantum cluster monomials, extending classical results to the quantum setting.

## Contribution

It constructs and analyzes quantum twist automorphisms, linking them to representation theory and quantum cluster algebras, and extends classical automorphism properties to the quantum context.

## Key findings

- Quantum twist automorphisms preserve dual canonical bases.
- Quantum twist automorphisms are described by syzygy functors for preprojective algebra representations.
- Quantum twist automorphisms are compatible with quantum cluster monomials.

## Abstract

In this paper, we construct twist automorphisms on quantum unipotent cells, which are quantum analogues of the Berenstein-Fomin-Zelevinsky twist automorphisms on unipotent cells. We show that those quantum twist automorphisms preserve the dual canonical bases of quantum unipotent cells. Moreover, we prove that quantum twist automorphisms are described by the syzygy functors for representations of preprojective algebras in the symmetric case. This is the quantum analogue of Gei{\ss}-Leclerc-Schr\"oer's description, and Gei{\ss}-Leclerc-Schr\"oer's results are essential in our proof. As a consequence, we show that quantum twist automorphisms are compatible with quantum cluster monomials. The 6-periodicity of specific quantum twist automorphisms is also verified.