Maximal green sequences for quantum and Poisson CGL extensions

TL;DR

This paper proves that all symmetric quantum and Poisson CGL extensions have maximal green sequences, extending previous explicit constructions to a broad class of cluster algebras.

Contribution

It establishes the existence of maximal green sequences for all symmetric quantum and Poisson CGL extensions, generalizing prior specific cases.

Findings

Maximal green sequences exist for all symmetric quantum CGL extensions.

Maximal green sequences exist for all symmetric Poisson CGL extensions.

Many previously known sequences are special cases of this general result.

Abstract

We prove that the quantum and classical cluster algebras for all members of the axiomatically defined classes of symmetric quantum and Poisson Cauchon-Goodearl-Letzter extensions possess maximal green sequences in the sense of Keller. Previously, maximal green sequences were constructed for explicit families of cluster algebras; many of those can be recovered from the general result for CGL extensions.