论文标题
变形簇泊松品种的量化
Quantization of deformed cluster Poisson varieties
论文作者
论文摘要
Fock和Goncharov描述了[FG09]中群集$ \ Mathcal {x} $ - 品种（也称为cluster Poisson品种）的量化。同时，在[BFMNC18]中引入了集群$ \ MATHCAL {x} $变形的家族。在本文中，我们表明这两个构造是兼容的 - 我们扩展了$ \ Mathcal {x} $的Fock-Goncharov量化 - [BFMNC18]的家族。作为推论，我们得到这些家庭及其每个纤维都有泊松结构。我们将这种结构与$ \ Mathcal {a} $ - 品种（[BZ05]）的Berenstein-Zelevinsky量化相关联。最后，灵感来自[LLRZ14]中量子贪婪基础的量子阳性的启发，我们计算了一个反例以量子theta基础的量子阳性。
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.
