Poisson and quantum geometry of acyclic cluster algebras

TL;DR

This paper explores the geometric and algebraic structures of acyclic cluster algebras, demonstrating their connection to holomorphic symplectic manifolds and analyzing properties of their quantum counterparts.

Contribution

It establishes that certain acyclic cluster algebras are coordinate rings of holomorphic symplectic manifolds and shows quantum cluster algebras lack non-trivial prime ideals, supporting a generalized orbit method.

Findings

Acyclic cluster algebras over complex numbers are coordinate rings of holomorphic symplectic manifolds.

Quantum cluster algebras associated with these have no non-trivial prime ideals.

Provides evidence for a generalized orbit method in quantized coordinate rings.

Abstract

We prove that certain acyclic cluster algebras over the complex numbers are the coordinate rings of holomorphic symplectic manifolds. We also show that the corresponding quantum cluster algebras have no non-trivial prime ideals. This allows us to give evidence for a generalization of the conjectured variant of the orbit method for quantized coordinate rings and their classical limits.