A. B. Goncharov
We express the Aomoto trilogarithm explicitely via classical trilogarithm and investigate the algebraic-geometric structures behind this: different realuzations of the weight three motivic complexes. Using this results we give an explicit motivic construction of the Grassmannian 4-logarithm and Borel regulator map for K_7(C).
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Some time ago the first author suggested two different constructions of Grassmannian n-logarithms. One of them works for all n, but its motivic nature is hidden. The other is a motivic but so far available only for n=2,3,4. We derived some general results about the first construction of Grassmannian polylogarithms, and found an explicit relation between the two constructions of the Grassmannian trilogarithms.
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 complet…
We associate motivic zeta functions to a large class of infinite dimensional Lie algebras
Consider the neutral Tannakian category mixed Tate motives over Z, in this paper we suggest a way to understand the structure of depth-graded motivic Lie subalgebra generated by the depth one part. We will show that from an isomorphism conjecture proposed by K. Tasaka we can deduce the F. Brown matrix conjecture and the non-degenerated conjecture about depth-graded motivic Lie subalgebra generated by the depth one part.
Relating depth graded and block graded motivic Lie algebras
本文以块滤过实现余根滤过，建立块分次与全奇多重 zeta 值之间的字典，刻画深度分次动机李代数关系，并探讨非均匀 Broadhurst–Kreimer 猜想。
We study the action of the inertia operator on the motivic Hall algebra, and prove that it is diagonalizable. This leads to a filtration of the Hall algebra, whose associated graded algebra is commutative. In particular, the degree 1 subspace forms a Lie algebra, which we call the Lie algebra of virtually indecomposable elements, following Joyce. We prove that the integral of virtually indecomposable elements admits an Euler characteristic specialization. In order to take advantage of the fact that our inertia groups are unit groups in algebras, we introduce the notion of algebroid.
The motivic Lie algebra embeds into the cohomology of the general linear group
该研究证明整数环上混合 Tate 动机的动机李代数可典范嵌入 GL_g(ℤ) 及 𝒜_g 的紧支上同调，通过热带几何等构造，结合相关理论得出图上同调对应动机李代数生成元，还计算了边数≤14 的图的典范图上同调。
Motivic coaction and single-valued map of polylogarithms from zeta generators
该研究提出一种新的李代数方法，通过奇zeta值相关李代数生成元共轭多重对数生成级数来控制多重zeta值，显式构造任意变量数多重对数的动机上作用与单值映射，保留纤维化基选择，揭示不同深度多重zeta值关联并为亏格零外推广铺路。
On a relation of a conjecture of Goncharov to the co-Lie algebra of Bloch-Kriz mixed Tate motives
研究通过混合塔克动机的co-Lie代数定义从Goncharov群B_n(F)到动机多项式对数的线性映射，支持Beilinson和Soulé关于字段K-群消失的猜想。
In this paper we describe a connection between realizations of the action of the motivic Galois group on the motivic fundamental groups of Gaussian and Eisenstein elliptic curves punctured at the $\mathfrak{p}$-torsion points, $π_1^{\rm Mot}(E-E[\mathfrak{p}],v_0)$, and the geometry of the Bianchi hyperbolic threefolds $Γ_1(\mathfrak{p})\setminus\mathbb{H}^3$, where $Γ_1(\mathfrak{p})$ is a congruence subgroup of ${\rm GL}_2({\rm End}(E))$. The first instance of such a connection was found by A.Goncharov (arXiv:math/0510310). In particular, we study the Hodge realization of the image of the ab…
Two semisimple algebraic groups are said to be motivic equivalent if the motives of the associated twisted flag varieties are isomorphic modulo any prime p. The purpose of this note is to construct the combinatorial invariants which characterize motivic equivalence and which are the motivic analogues of the Tits indices which appear in the classification of semisimple algebraic groups. The expression of these invariants -the Tits p-indexes- in terms of the classical invariants associated to the natural underlying structures of semisimple algebraic groups allow to produce algebraic criteria of …
Motivic correlators, cluster varieties and Zagier's conjecture on zeta(F,4)
研究证明数域F的戴德金ζ函数在s = 4处取值的扎吉尔猜想，通过定义特定映射等工具，还给出椭圆曲线E的L函数在s = 4处值的公式，为自由性猜想中F的动机李余代数权重4部分提供证据。
This text gives a construction of a differential graded Lie algebra in Nori's category of effective homological motives. In fact the construction works in more a general setting than that of an Abelian category. This allows us to give the rational homotopy Lie algebra of a 1-connected space a motivic structure. As a consequence the rational homotopy Lie algebra inherits a mixed Hodge structure and Galois module structure.
We define reduced zeta functions of Lie algebras, which can be derived from motivic zeta functions using the Euler characteristic. We show that reduced zeta functions of Lie algebras possessing a suitably well-behaved basis are easy to analyse. We prove that reduced zeta functions are multiplicative under certain conditions and investigate which reduced zeta functions have functional equations.
Motivic homotopy theory of the classifying stack of finite groups of Lie type
本文将Lie型有限群分类栈的Tate动机与旗流形的等变Tate动机相关联，将Edidin-Graham的经典结果推广至高阶等变Chow群，并计算了G-zips栈的动机。
We prove that the degree zero Hochschild-type cohomology of the homology operad of Francis Brown's dihedral moduli spaces is equal to the Grothendieck-Teichmueller Lie algebra plus two classes. This significantly elucidates the (in part still conjectural) relation between the Grothendieck-Teichmueller Lie algebra and (motivic) multiple zeta values.
For a given symmetric quiver $Q$, we define a supercommutative quadratic algebra $\mathcal{A}_Q$ whose Poincaré series is related to the motivic generating function of $Q$ by a simple change of variables. The Koszul duality between supercommutative algebras and Lie superalgebras assigns to the algebra $\mathcal{A}_Q$ its Koszul dual Lie superalgebra $\mathfrak{g}_Q$. We prove that the motivic Donaldson-Thomas invariants of the quiver $Q$ may be computed using the Poincaré series of a certain Lie subalgebra of $\mathfrak{g}_Q$ that can be described, using an action of the first Weyl algebra on …
We study the action of the Galois group on the pro-l-completion of the fundamental group of P^1 - {0, infinity and N-th roots of unity}. We describe the Lie algebra of the image of the Galois action and relate with the geometry of the modular varieties for GL_m for m = 1,2,3,... This story is the l-adic side of the motivic theory of multiple polylogarithms at roots of unity, which generalize the classical cyclotomy theory.
Let M be a 1-motive defined over a field of characteristic 0. To M we can associate its motivic Galois group, G_mot(M), which is the geometrical interpretation of the Munford-Tate group of M. We prove that the unipotent radical of the Lie algebra of G_mot(M) is the semi-abelian variety defined by the adjoint action of the semi-simplified of the Lie algebra of G_mot(M) on itself.
Cellular $\mathbb A^1$-homology and the motivic version of Matsumoto's theorem
本文定义了适用于域上光滑概形的胞腔 $\mathbb A^1$-同调，通过显式计算确定了任意域上分裂约化群的 $\mathbb A^1$-基本群，从而获得了 Matsumoto 定理的动形版本，并统一推广了相关的中心扩张与同伦群结果。
We investigate the nature of divergences in quantum field theory, showing that they are organized in the structure of a certain `` motivic Galois group'', which is uniquely determined and universal with respect to the set of physical theories. The renormalization group can be identified canonically with a one parameter subgroup. The group is obtained through a Riemann-Hilbert correspondence. Its representations classify equisingular flat vector bundles, where the equisingularity condition is a geometric formulation of the fact that in quantum field theory the counterterms are independent of th…
The nilpotent bicone of a finite dimensional complex reductive Lie algebra g is the subset of elements in g x g whose subspace generated by the components is contained in the nilpotent cone of g. The main result of this note is that the nilpotent bicone is a complete intersection. This affirmatively answers a conjecture of Kraft-Wallach concerning the nullcone. In addition, we introduce and study the characteristic submodule of g. The properties of the nilpotent bicone and the characteristic submodule are known to be very important for the understanding of the commuting variety and its ideal o…
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