2020/07/06 by Man-Wai Cheung, Cheung, Man-Wai Mandy, Juan Bosco Frías-Medina +3
Mathematics · #13F60 (Primary) 14D06 #81R60 (Secondary) #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.2007.02479
openalex publication_date 2020/07/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Fock and Goncharov described a quantization of cluster X-varieties (also known as cluster Poisson varieties) in [FG09]. Meanwhile, families of deformations of cluster X-varieties were introduced in [BFMNC18]. In this paper we show that the two constructions are compatible -- we extend the Fock-Goncharov quantization of X-varieties to the families of [BFMNC18]. 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 ([BZ05]). Finally, inspired by the counter-example to quantum positivity of the quantum greedy basis in [LLRZ14], we compute a counter-example to quantum positivity of the quantum theta basis.