Poisson and quantum geometry of acyclic cluster algebras

2012/10/22 by Zwicknagl, Sebastian
#13F60 #16T20 #17B63 #FOS: Mathematics #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.1210.5824

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.

Related

• Cluster Algebras, Symplectic Leaves and Quantum Groups
• Poisson Ideals in Cluster Algebras and the Spectra of Quantized Coordinate Rings
• Toric Poisson Ideals in Cluster Algebras
• Cluster algebra structures on Poisson nilpotent algebras
• Quantum cluster algebras