Quantization of deformed cluster Poisson varieties
Abstract
Fock and Goncharov described a quantization of cluster -varieties (also known as cluster Poisson varieties) in [FG09]. Meanwhile, families of deformations of cluster -varieties were introduced in [BFMNC18]. In this paper we show that the two constructions are compatible -- we extend the Fock-Goncharov quantization of -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 -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.
Cite
@article{arxiv.2007.02479,
title = {Quantization of deformed cluster Poisson varieties},
author = {Man-Wai Mandy Cheung and Juan Bosco Frías-Medina and Timothy Magee},
journal= {arXiv preprint arXiv:2007.02479},
year = {2023}
}
Comments
46 pages, 4 figures, 2 tables. We added Definition 3.5 in order to obtain a correct proof of Proposition 3.6 (formerly Proposition 3.5). We added a discussion of Definition 3.11 (formerly Definition 3.10) in which we describe the gluings in greater detail. Other minor corrections. To appear in Algebras and Representation Theory