arXiv 2003.065212020-03-14math.AG

Hodge和motivic算子的混洗关系

Shuffle relations for Hodge and motivic correlators

Nikolay Malkin

备注：: 52 pages, 14 figures

摘要：

霍奇算子${\rm Cor}_{\mathcal H}(z_0,z_1,\dots,z_n)$是由Goncharov（arXiv：0803.0297）通过显式积分公式定义的多个复变量的函数。它们满足一些线性关系：二面角对称关系、分布关系和混洗关系。我们发现了新的第二次洗牌关系。当$z_i\in0\cupμ_N$，其中$μ_N$是$N$ -th单位根，他们预计将提供几乎所有的关系。当$z_i$遍历$\mathbb C$的有限子集$S$时，Hodge算子描述了在基本群$π_1^{\rm nil}(\mathbb{CP}^1-(S\cup\infty),v_\infty)$的幂零完备上的实混合Hodge-Tate结构，基本群是混合$\mathbb Q$ -Hodge-Tate结构范畴中的李代数. Hodge算子被提升为该范畴的Tannakian李余代数中的标准元${\rm Cor_{Hod}}(z_0,\dots,z_n)$.我们证明了这些元素满足第二个洗牌关系。令$S\subset\overline{\mathbb Q}$。幂零基本群是motivic基本群的Betti实现，motivic基本群是$\overline{\mathbb Q}$上的混合Tate motives范畴中的李代数。将Hodge算子提升到其Tannakian李余代数$\rm Lie_{MT}^\vee$中的元素${\rm Cor_{Mot}}(z_0,\dots,z_n)$。我们证明了这些动机元素的第二个洗牌关系。$\rm Lie_{MT}^\vee$的泛包络代数是由Goncharov通过motivic多重多项式来描述的，它们服从一组相似但不同的双混洗关系。动机交换子有几个优点：它们在所有点上都服从二面角对称关系，而不仅仅是在单位根上;它们是为任何曲线定义的，双洗牌关系允许推广到椭圆曲线;它们描述动机李余代数的元素，而不是它的泛包络代数。

英文摘要：

The Hodge correlators ${\rm Cor}_{\mathcal H}(z_0,z_1,\dots,z_n)$ are functions of several complex variables, defined by Goncharov (arXiv:0803.0297) by an explicit integral formula. They satisfy some linear relations: dihedral symmetry relations, distribution relations, and shuffle relations. We found new second shuffle relations. When $z_i\in0\cupμ_N$, where $μ_N$ are the $N$-th roots of unity, they are expected to give almost all relations. When $z_i$ run through a finite subset $S$ of $\mathbb C$, the Hodge correlators describe the real mixed Hodge-Tate structure on the pronilpotent completion of the fundamental group $π_1^{\rm nil}(\mathbb{CP}^1-(S\cup\infty),v_\infty)$, a Lie algebra in the category of mixed $\mathbb Q$-Hodge-Tate structures. The Hodge correlators are lifted to canonical elements ${\rm Cor_{Hod}}(z_0,\dots,z_n)$ in the Tannakian Lie coalgebra of this category. We prove that these elements satisfy the second shuffle relations. Let $S\subset\overline{\mathbb Q}$. The pronilpotent fundamental group is the Betti realization of the motivic fundamental group, a Lie algebra in the category of mixed Tate motives over $\overline{\mathbb Q}$. The Hodge correlators are lifted to elements ${\rm Cor_{Mot}}(z_0,\dots,z_n)$ in its Tannakian Lie coalgebra $\rm Lie_{MT}^\vee$. We prove the second shuffle relations for these motivic elements. The universal enveloping algebra of $\rm Lie_{MT}^\vee$ was described by Goncharov via motivic multiple polylogarithms, which obey a similar yet different set of double shuffle relations. Motivic correlators have several advantages: they obey dihedral symmetry relations at all points, not only at roots of unity; they are defined for any curve, and the double shuffle relations admit a generalization to elliptic curves; and they describe elements of the motivic Lie coalgebra rather than its universal enveloping algebra.

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