Quasi-reflection algebras and cyclotomic associators
Quasi-reflection algebras and cyclotomic associators
复制标题
准反射代数和分圆关联子
DOI：
10.1007/s00029-007-0048-2
复制
发表时间：
2004
期刊：
影响因子：
--
通讯作者：
B. Enriquez
中科院分区：
文献类型：
--
作者：
B. Enriquez
摘要：我们发展了结合子理论的割圆类比。利用泛函KZ方程的三角形式，我们证明了态射的形式
$$B_n^1\right tarrow({\mathbb{Z}}/N{\mathbb{Z}})^n\r次{\mathfrak{S}}_n$$
其中Bn1是B型辫子群。形式同构在代数上依赖于一系列ΨKZ，即“KZ伪扭曲”。我们研究了伪扭方案，证明了它是映射到Drinfeld群GT(K)的群GTM(N，k)下的挠子，并且它的李代数同构于它的相关分次群
$$\mathfrak{grtm}$$
(n，k)。我们证明了用分布关系定义的Grothendieck-Teichmüller群的Ihara子群GTK实际上与它重合。证明了满足分布关系的伪扭的子格式是次扭子格式。我们研究了相应的类比
$$\mathfrak{grtmd}$$
第(n，k)个
$$\mathfrak{grtm}$$
(n，k)；它是作用为的分次李代数
$$({\mathbb{Z}}/N{\mathbb{Z}})^{\次}$$
，并给出了它的生成空间特征的一个下界。
Abstract.We develop a cyclotomic analogue of the theory of associators. Using a trigonometric version of the universal KZ equations, we prove the formality of a morphism
$$B_n^1 \rightarrow ({\mathbb{Z}}/N{\mathbb{Z}})^n \rtimes {\mathfrak{S}}_n$$
, where Bn1 is a braid group of type B. The formality isomorphism depends algebraically on a series ΨKZ, the “KZ pseudotwist”. We study the scheme of pseudotwists and show that it is a torsor under a group GTM(N, k), mapping to Drinfeld’s group GT(k), and whose Lie algebra is isomorphic to its associated graded
$$\mathfrak{grtm}$$
(N, k). We prove that Ihara’s subgroup GTK of the Grothendieck–Teichmüller group, defined using distribution relations, in fact coincides with it. We show that the subscheme of pseudotwists satisfying distribution relations is a subtorsor. We study the corresponding analogue
$$\mathfrak{grtmd}$$
(N, k) of
$$\mathfrak{grtm}$$
(N, k); it is a graded Lie algebra with an action of
$$({\mathbb{Z}}/N{\mathbb{Z}})^{\times}$$
, and we give a lower bound for the character of its space of generators.