Cluster Algebras, Symplectic Leaves and Quantum Groups

TL;DR

This paper explores the Poisson geometry of cluster algebras and their quantum analogs, providing new insights into their ideal structures and a novel construction related to the Dixmier map.

Contribution

It introduces a new approach to the ideal theory of quantum cluster algebras and a novel construction of the Dixmier map connecting symplectic leaves and primitive ideals.

Findings

Established a connection between symplectic leaves and primitive ideals.

Provided evidence supporting the homeomorphism of the Dixmier map.

Applied the approach to quantized coordinate rings.

Abstract

This paper investigates the Poisson geometry of cluster algebras and the corresponding ideal theory of quantum cluster algebras. We then show how our approach can be applied to the ring theory of quantized coordinate rings. We give a new construction for the Dixmier map constructed by Yakimov from the space of symplectic leaves on $\CC[G]$ to the space of primitive ideals on $\CC_q[G]$ and give further evidence that this map is a homeomorphism.