Poisson geometry and Azumaya loci of cluster algebras. (English) Zbl 07915892
Cluster algebras were defined by S. Fomin and A. Zelevinsky [J. Am. Math. Soc. 15, No. 2, 497–529 (2002; Zbl 1021.16017)], and since then, they have played prominent role in many areas of mathematics and mathematical physics. The upper cluster algebras \(\mathsf{U}\) with their Gekhtman–Shapiro–Vainshtein (GSV) Poisson brackets and their root of unity quantizations \(\mathsf{U}_{\varepsilon}\) are the main types of objects in the theory of cluster algebras. On the Poisson side, the authors prove that the spectrum of every finitely generated upper cluster algebra \(\mathsf{U}\) with its GSV Poisson structure always has a Zariski open orbit of symplectic leaves and give an explicit description of it. On the quantum side, the authors describe the fully Azumaya loci of the quantizations \(\mathsf{U}_{\varepsilon}\) under the assumption that \(\mathsf{A}_{\varepsilon}=\mathsf{U}_{\varepsilon}\) and \(\mathsf{U}_{\varepsilon}\) is a finitely generated algebra. All results allow frozen variables to be either inverted or not.
Reviewer: Mee Seong Im (Annapolis)
MSC:
| 13F60 | Cluster algebras |
| 53D17 | Poisson manifolds; Poisson groupoids and algebroids |
| 16G30 | Representations of orders, lattices, algebras over commutative rings |
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
Keywords:
cluster algebras; Gekhtman-Shapiro-Vainshtein Poisson brackets; torus orbits of symplectic leaves; root of unity quantum cluster algebras; fully Azumaya loci; Poisson ordersCitations:
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