Quantization of Deformed Cluster Poisson Varieties

Fock and Goncharov described a quantization of cluster \(\mathcal {X}\)-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 \(\mathcal {X}\)-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 \(\mathcal {X}\)-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 \(\mathcal {A}\)-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.