Twist automorphisms and Poisson structures

TL;DR

This paper introduces quantum twist automorphisms for cluster algebras and Poisson structures, generalizing previous automorphisms and exploring their properties, compatibility, and explicit constructions.

Contribution

It presents a general framework for twist automorphisms in cluster algebras, extending known automorphisms and providing explicit examples for quantum and principal coefficient cases.

Findings

Twist automorphisms permute well-behaved bases.

Existence and compatibility with Poisson structures established.

Explicit constructions for Donaldson-Thomas type automorphisms.

Abstract

We introduce (quantum) twist automorphisms for upper cluster algebras and cluster Poisson algebras with coefficients. Our constructions generalize the twist automorphisms for quantum unipotent cells. We study their existence and their compatibility with Poisson structures and quantization. The twist automorphisms always permute well-behaved bases for cluster algebras. We explicitly construct (quantum) twist automorphisms of Donaldson-Thomas type and for principal coefficients.