B_4$ 的布劳表示和有理投影面的量子化

Perrine Jouteur
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引用次数: 0

摘要

辫子群 $B_4$ 自然地作用于有理投影面$\mathbb{P}^2(\mathbb{Q})$,这个作用对应于 $B_4$ 的经典积分还原布劳表示。本文的第一个结果是对这一作用的轨道进行分类。布劳表示定义了 $B_4$ 在 $\mathbb{P}^2(\mathbb{Z}(q))$ 上的作用,其中 $q$ 是形式参数,$\mathbb{Z}(q)$ 是在 $q$ 中具有整数系数的有理函数域。我们研究了$B_4$作用在$mathbb{P}^2(\mathbb{Z}(q))$上的轨道,并证明了$q$变形投影线$mathbb{P}^1(\mathbb{Z}(q))$的嵌入的存在,它精确地对应于莫里埃-杰努德(Morier-Genoud)和奥夫先科(Ovsienko)提出的$q$有理概念。
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Burau representation of $B_4$ and quantization of the rational projective plane
The braid group $B_4$ naturally acts on the rational projective plane $\mathbb{P}^2(\mathbb{Q})$, this action corresponds to the classical integral reduced Burau representation of $B_4$. The first result of this paper is a classification of the orbits of this action. The Burau representation then defines an action of $B_4$ on $\mathbb{P}^2(\mathbb{Z}(q))$, where $q$ is a formal parameter and $\mathbb{Z}(q)$ is the field of rational functions in $q$ with integer coefficients. We study orbits of the $B_4$-action on $\mathbb{P}^2(\mathbb{Z}(q))$, and show existence of embeddings of the $q$-deformed projective line $\mathbb{P}^1(\mathbb{Z}(q))$ that precisely correspond to the notion of $q$-rationals due to Morier-Genoud and Ovsienko.
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