Quantization of Deformed Cluster Poisson Varieties

• Published in: Algebras and representation theory Vol. 27; no. 1; pp. 381 - 427
• Main Authors: Cheung, Man-Wai Mandy, Frías-Medina, Juan Bosco, Magee, Timothy
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.02.2024; Springer Nature B.V
• Subjects: Associative Rings and Algebras; Clusters; Commutative Rings and Algebras; Deformation; Mathematics; Mathematics and Statistics; Non-associative Rings and Algebras; Cluster varieties; Quantum cluster algebras; Poisson structure; Toric degenerations
• ISSN: 1386-923X; 1572-9079
• DOI: 10.1007/s10468-023-10209-x

Fock and Goncharov described a quantization of cluster X -varieties (also known as cluster Poisson varieties ) in Fock and Goncharov (Ann. Sci. Éc. Norm. Supér. 42 (6), 865–930 2009 ). Meanwhile, families of deformations of cluster X -varieties were introduced in Bossinger et al. (Compos. Math. 156 (10), 2149–2206, 2020 ). In this paper we show that the two constructions are compatible– we extend the Fock-Goncharov quantization of X -varieties to the families of Bossinger et al. (Compos. Math. 156 (10), 2149–2206, 2020 ). As a corollary, we obtain that these families and each of their fibers have Poisson structures. We relate this construction to the Berenstein-Zelevinsky quantization of A -varieties (Berenstein and Zelevinsky, Adv. Math. 195 (2), 405–455, 2005 ). Finally, inspired by the counter-example to quantum positivity of the quantum greedy basis in Lee, et al. (Proc. Natl. Acad. Sci. 111 (27), 9712–9716, 2014 ), we compute a counter-example to quantum positivity of the quantum theta basis.