Dimitri Tamarkin
Action of the graded Grothendieck-Teichmueller (GT) group on a resolution of the operad of Gerstenhaber algebras (GA) is defined. It is shown that the induced Lie algebra action is homotopically non-trivial (i.e. the induced map from the Lie algebra of the graded GT group to the deformation complex of the operad of GA is injective).
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This paper attempts to provide a more or less self-contained introduction into theory of the Grothendieck-Teichmueller group and Drinfeld associators using the theory of operads and graph complexes.
We give the operadic formulation of (weak, strong) topological vertex algebras, which are variants of topological vertex operator algebras studied recently by Lian and Zuckerman. As an application, we obtain a conceptual and geometric construction of the Batalin-Vilkovisky algebraic structure (or the Gerstenhaber algebra structure) on the cohomology of a topological vertex algebra (or of a weak topological vertex algebra) by combining this operadic formulation with a theorem of Getzler (or of Cohen) which formulates Batalin-Vilkovisky algebras (or Gerstenhaber algebras) in terms of the homolog…
We show that the zeroth cohomology of M. Kontsevich's graph complex is isomorphic to the Grothendieck-Teichmueller Lie algebra grt_1. The map is explicitly described. This result has applications to deformation quantization and Duflo theory. We also compute the homotopy derivations of the Gerstenhaber operad. They are parameterized by grt_1, up to one class (or two, depending on the definitions). More generally, the homotopy derivations of the (non-unital) E_n operads may be expressed through the cohomology of a suitable graph complex. Our methods also give a second proof of a result of H. Fur…
In this article, we show under what additional ingredients a comp (or opposite) module over an operad with multiplication can be given the structure of a cyclic k-module and how the underlying simplicial homology gives rise to a Batalin-Vilkovisky module over the cohomology of the operad. In particular, one obtains a generalised Lie derivative and a generalised (cyclic) cap product that obey a Cartan-Rinehart homotopy formula, and hence yield the structure of a noncommutative differential calculus in the sense of Nest, Tamarkin, Tsygan, and others. Examples include the calculi known for the Ho…
This paper is concerned by the concept of algebra up to homotopy for a structure defined by two operations $.$ and [,]. An important example of such a structure is the Gerstenhaber algebra (commutatitve and Lie). The notion of Gerstenhaber algebra up to homotopy ($G_\infty$ algebra) is known. Here, we give a definition of pre-Gerstenhaber algebra (pre-commutative and pre-Lie) allowing the construction of $\hbox{pre}G_\infty$ algebra. Given a structure of pre-commutative (Zinbiel) and pre-Lie algebra and working over the corresponding dual operads, we will give an explicit construction of the a…
This paper is a little more detailed version of math-QA/0010017 "Sur l'homologie des espaces de nœuds non-compacts", where the first term of the Vassiliev spectral sequence (computing the homology of the space of long knots in ${\mathbb R}^d$, $d\ge 3$) was described in terms of the Hochschild homology of the Poisson algebras operad for $d$ odd, and of the Gerstenhaber algebras operad for $d$ even. In particular, the bialgebra of chord diagrams arises as some subspace of this homology. The homology in question is the space of characteristic classes for Hochschild cohomology of Poisson (resp. G…
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.
We show that the diagonal complex computing the Gerstenhaber-Schack cohomology of a bialgebra (that is, the cohomology theory governing bialgebra deformations) can be given the structure of an operad with multiplication if the bialgebra is a (not necessarily finite dimensional) Hopf algebra with invertible antipode; if the antipode is involutive, the operad is even cyclic. Therefore, the Gerstenhaber-Schack cohomology of any such Hopf algebra carries a Gerstenhaber resp. Batalin-Vilkovisky algebra structure; in particular, one obtains a cup product and a cyclic boundary B that generate the Ger…
It is clarified how cohomologies and Gerstenhaber algebras can be associated with linear pre-operads (comp algebras). Their relation to mechanics and operadic physics is concisely discussed.
Using non-commutative differential forms, we construct a complex called singular Hochschild cochain complex for any associative algebra over a field. The cohomology of this complex is isomorphic to the Tate-Hochschild cohomology in the sense of Buchweitz. By a natural action of the cellular chain operad of the spineless cacti operad, introduced by R. Kaufmann, on the singular Hochschild cochain complex, we provide a proof of the Deligne's conjecture for this complex. More concretely, the complex is an algebra over the (dg) operad of chains of the little $2$-discs operad. By this action, we als…
The essential parts of the operad algebra are concisely presented, which should be useful when confronting with the operadic physics. It is also clarified how the Gerstenhaber algebras can be associated with the linear pre-operads (comp algebras). Their relation to mechanics is concisely discussed. A hypothesis that the Feynman diagrams are observables is proposed.
The goal of this paper is to complete Getzler-Jones' proof of Deligne's Conjecture, thereby establishing an explicit relationship between the geometry of configurations of points in the plane and the Hochschild complex of an associative algebra. More concretely, it is shown that the $B_\infty$-operad, which is generated by multilinear operations known to act on the Hochschild complex, is a quotient of a certain operad associated to the compactified configuration spaces. Different notions of homotopy Gerstenhaber algebras are discussed: one of them is a $B_\infty$-algebra, another, called a hom…
We define a generalization of Massey products for algebras over a Koszul operad in characteristic zero, extending Massey's and Allday's and Retah's in the associative and Lie cases, respectively. We establish connections with minimal models and with Dimitrova's universal operadic cohomology class. We compute a Gerstenhaber algebra example and a hypercommutative algebra example related to the Chevalley-Eilenberg complex of the Heisenberg Lie algebra.
The operads of Poisson and Gerstenhaber algebras are generated by a single binary element if we consider them as Hopf operads (i.e. as operads in the category of cocommutative coalgebras). In this note we discuss in details the Hopf operads generated by a single element of arbitrary arity. We explain why the dual space to the space of $n$-ary operations in this operads are quadratic and Koszul algebras. We give the detailed description of generators, relations and a certain monomial basis in these algebras.
We prove that the BRST complex of a topological conformal field theory is a homotopy Gerstenhaber algebra, as conjectured by Lian and Zuckerman in 1992. We also suggest a refinement of the original conjecture for topological vertex operator algebras. We illustrate the usefulness of our main tools, operads and "string vertices" by obtaining new results on Vassiliev invariants of knots and double loop spaces.
The solution of Deligne's conjecture on Hochschild cochains and the formality of the operad of little disks provide us with a natural homotopy Gerstenhaber algebra structure on the Hochschild cochains of an associative algebra. In this paper we construct a natural chain of quasi-isomorphisms of homotopy Gerstenhaber algebras between the Hochschild cochain complex C(A) of a regular commutative algebra A over a field of characteristic zero and the Gerstenhaber algebra of multiderivations of A. Unlike the original approach of the second author based on the computation of obstructions our method a…
Homotopy G-algebras and moduli space operad
本文强调了代数曲线模空间在结合代数与拓扑中的作用，证明了带乘法operad空间构成同伦Gerstenhaber代数，奇异上链复形自然构成operad，且装饰模空间operad在Kähler流形的de Rham复形上诱导出最一般的同伦G-代数结构。
The spectral sequence constructed by V.A.Vassiliev computes the homology of the spaces of non-compact knots in ${\bf R}^d$, $d\ge 3$. In this work the first term of this spectral sequence is described in terms of the homology of the Hochschild complex for the Poisson algebras operad, if d is odd (resp. for the Gerstenhaber algebras operads, if d is even). In particular the bialgebra of chord diagrams arises as some subspace of this homology (in this case d=3). Also a simplification for the calculation of the Vassiliev spectral sequence in the first term is provided.
Cyclic duality between BV algebras and BV modules
本文证明满足cosimplicial相容条件的循环operad其上同调具有BV代数结构，并给出BV模的循环对偶成为BV代数的条件，以Hopf代数理论中的同调群为例加以说明。
Higher structures on homology groups
本文将 operad 乘法诱导 Gerstenhaber 代数的经典结论对偶到 cooperad，证明带余乘法的 cooperad 赋予同调群 Gerstenhaber 余代数结构，并应用于 Tor 群与 Hochschild 同调。
The abelian category of tetramodules over an associative bialgebra $A$ is related with the Gerstenhaber-Schack (GS) cohomology as $Ext_\Tetra(A,A)=H_\GS(A)$. We construct a 2-fold monoidal structure on the category of tetramodules of a bialgebra. Suppose $C$ is an abelian $n$-fold monoidal category with the unit object $A$. We prove, provided some condition (*), that $Ext_C(A,A)$ is an $(n+1)$-algebra. In the case of bialgebras this condition (*) is satisfied when $A$ is a Hopf algebra. Finally, the GS cohomology of a Hopf algebra is a 3-algebra. As well, we consider this kind of questions of …
Semi-infinite forms on the moduli spaces of genus-zero Riemann surfaces with punctures and local coordinates are introduced. A partial operad for semi-infinite forms is constructed. Using semi-infinite forms and motivated by a partial suboperad of the partial operad for semi-infinite forms, topological vertex partial operads of type $k<0$ and strong topological vertex partial operads of type $k<0$ are constructed. It is proved that the category of (locally-)grading-restricted (strong) topological vertex operator algebras of type $k<0$ and the category of (weakly) meromorphic ${\Bbb Z}\times {\…
In this note, we show that the normalized Hochschild co--chains of an associative algebra with a non--degenerate, symmetric, invariant inner product are an algebra over a chain model of the framed little discs operad which is given by cacti. In particular, in this sense they are a BV algebra up to homotopy and the Hochschild cohomology of such an algebra is a BV algebra whose induced bracket coincides with Gerstenhaber's bracket. To show this, we use a cellular chain model for the framed little disc operad in terms of normalized cacti. This model is given by tensoring our chain model for the l…
We study Thomas Willwacher's twisting endofunctor tw in the category of dg properads P under the operad of (strongly homotopy) Lie algebras. It is proven that if P is a properad under properad Lieb of Lie bialgebras , then the associated twisted properad tw(P) becomes in general a properad under quasi-Lie bialgebras (rather than under Lieb). This result implies that the cyclic cohomology of any cyclic homotopy associative algebra has in general an induced structure of a quasi-Lie bialgebra. We show that the cohomology of the twisted properad tw(Lieb) is highly non-trivial -- it contains the co…
Brackets and products from centres in extension categories
该研究将Schwede关于结合代数双模上Ext群的Gerstenhaber代数构造推广至带幺半中心系数的一般幺半范畴，显式刻画了双代数胚左模范畴情形下的结构并给出operad证明，推广了经典结果，同时提出更一般情形的猜想。
We compute the Poincare polynomial and the cohomology algebra with rational coefficeints of the manifold M_n of real points of the moduli space of algebraic curves of genus 0 with n labeled points. This cohomology is a quadratic algebra, and we conjecture that it is Koszul. We also compute the 2-local torsion in the cohomology of M_n. As was shown by E. Rains in arXiv:math/0610743 the cohomology of M_n does not have odd torsion, so that the above determines the additive structure of the integral homology and cohomology. Further, we prove that the rational homology operad of M_n is the operad o…
In two seminal papers Kontsevich used a construction called_graph homology_ as a bridge between certain infinite dimensional Lie algebras and various topological objects, including moduli spaces of curves, the group of outer automorphisms of a free group, and invariants of odd dimensional manifolds. In this paper, we show that Kontsevich's graph complexes, which include graph complexes studied earlier by Culler and Vogtmann and by Penner, have a rich algebraic structure. We define a Lie bracket and cobracket on graph complexes, and in fact show that they are Batalin-Vilkovisky algebras, and th…
Obstruction theory for $A$-infinity bimodules
本文构建截断最小 $A$-无穷双模扩张的障碍理论，以新型边缘谱序列和上同调刻画障碍，并推出微分分次双模的内在形式性与近乎形式性结果。
Brace $B_{\infty}$ algebras associated with Hopf algebroids
本研究将brace $B_{\u221e}$代数的operadic建模应用于Xu型Hopf代数胚，构造了I型与II型扭brace $B_{\u221e}$代数间的严格$B_{\u221e}$同构，并借此揭示了两类源自李代数对的brace $B_{\u221e}$代数间的隐藏关联。
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