Quantization of deformed cluster Poisson varieties

TL;DR

This paper extends the Fock-Goncharov quantization to deformed families of cluster $ ext{X}$-varieties, establishing their Poisson structures and relating them to Berenstein-Zelevinsky quantization, while also providing a counter-example to quantum positivity.

Contribution

It demonstrates the compatibility of quantization with deformations of cluster $ ext{X}$-varieties and connects this to existing quantizations, also addressing quantum positivity issues.

Findings

Extended Fock-Goncharov quantization to deformation families.

Established Poisson structures on these families and fibers.

Provided a counter-example to quantum positivity of the quantum theta basis.

Abstract

Fock and Goncharov described a quantization of cluster $\mathcal{X}$-varieties (also known as cluster Poisson varieties) in [FG09]. Meanwhile, families of deformations of cluster $\mathcal{X}$-varieties were introduced in [BFMNC18]. 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 [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 $\mathcal{A}$-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.