Quadratic algebras related to the bihamiltonian operad
Mikhail Bershtein, Vladimir Dotsenko, Anton Khoroshkin
本文证明了 B.L. Feigin 关于某些二次代数维数与特征的猜想，并揭示了这些代数与双哈密顿算子代数分量对偶之间的自然同构关系。
我们证明了 B.L. Feigin 提出的关于某些二次代数的维数与特征的猜想。结果表明，这些代数自然同构于双哈密顿算子代数各分量的对偶。
We prove the conjectures on dimensions and characters of some quadratic algebras stated by B$.$L$.$Feigin. It turns out that these algebras are naturally isomorphic to the duals of the components of the bihamiltonian operad.
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We describe those binary quadratic operads generated by a two-dimensional space that are isomorphic to their Koszul dual operads.
We describe the Koszul dual of two quadratic operads on planar forests introduced to study the infinitesimal Hopf algebra of planar rooted trees and prove that these operads are Koszul.
Nonsymmetric versions of binary quadratic operads
本文研究了结合操作元与二元二次操作元的White Manin积，引入了非对称版本的概念，并给出了判断该积操作元具有此性质的准则，通过实例和反例展示了构造方法，并描述了某些允许非对称版本的操作元的组合性质。
The aim of this paper is to present remarkable classes of Lie-admissible algebras containing in particular the associative algebras, the Vinberg algebras and pre-Lie algebras. We determine the associated quadratic operads and their dual operads.
On the Dong Property for a binary quadratic operad
本文研究了二元非结合代数满足Dong引理的条件，证明Novikov和Novikov-Poisson代数满足该引理，并通过操作代数语言确定了Dong引理成立的操作代数类型。
Quadratic nonsymmetric quaternary operads
本研究从代数操作代数理论视角出发，运用计算线性代数与交换代数，探究二次非对称四元运算的关系空间，重点确定三次推论具最小或最大秩的二次关系。
The notion of a coherent unit action on algebraic operads was first introduced by Loday for binary quadratic nonsymmetric operads and generalized by Holtkamp, to ensure that the free objects of the operads carry a Hopf algebra structure. There was also a classification of such operads in the binary quadratic nonsymmetric case. We generalize the notion of coherent unit action to braided operads and show that free objects of braided operads with such an action carries a braided Hopf algebra structure. Under the conditions of binary, quadratic and nonsymmetric, we give a characterization and clas…
Various compatibility conditions among replicated copies of operations in a given algebraic structure have appeared in broad contexts in recent years. Taking an uniform approach, this paper gives an operadic study of compatibility conditions for nonsymmetric operads with unary and binary operations, and homogeneous quadratic and cubic relations. This generalizes the previous studies for binary quadratic operads. We consider three compatibility conditions, namely the linear compatibility, matching compatibility and total compatibility, with increasingly strict restraints among the replicated co…
We consider nonsymmetric operads with two binary operations satisfying relations in arity 3; hence these operads are quadratic, and so we can investigate Koszul duality. We first consider operations which are nonassociative (not necessarily associative) and then specialize to the associative case. We obtain a complete classification of self-dual quadratic nonsymmetric operads with two (associative or nonassociative) binary operations. These operads generalize associativity for one operation to the setting of two operations.
In the theory of binary quadratic operads, the white and black products of operads (called Manin products) play an important role. Given two such operads, the computation of either of their Manin products is a routine task. We present and describe a computer program that helps to compute white and black Manin products of binary quadratic operads. The same utility allows to find the Koszul-dual operads. In particular, we compute the white product of the operads Lie and As (governing the varieties of Lie and associative algebras, respectively). It turns out that the resulting operad is magmatic,…
Since its introduction by Loday in 1995 with motivation from algebraic K-theory, dendriform dialgebras have been studied quite extensively with connections to several areas in mathematics and physics. A few more similar structures have been found recently, such as the tri-, quadri-, ennea- and octo-algebras, with increasing complexity in their constructions and properties. We consider these constructions as operads and their products and duals, in terms of generators and relations, with the goal to clarify and simplify the process of obtaining new algebra structures from known structures and f…
Initial pre-algebras as a generalization of dendriform algebras
研究初始预代数，对二元运算律$\Var$定义初始预$\Var$ - 代数及运算律$\pre\Var^{\I}$，给出求定义关系的算法，研究自由初始预代数，还为非对称运算律$\pre\As^{\I}$构造基、描述线性基并建立双射。
Let P be a quadratic operad. We determine an associated operad ~P such that for any P-algebra A and any ~P-algebra B then the tensor product $A \otimes B$ is a P-algebra.
$δ$-Mock-Novikov Algebras: Structural Theory, Operad and Poisson-Type Constructions
本文引入参数依赖的 $\delta$-mock-Novikov 代数，研究其结构、operad 性质及 Poisson 型构造，证明有限维情形在特征零下幂零并完成低维分类。
The Weight Criterion, Dong Property, and Manin Products of Operads
本文研究二元 operad 的 Hadamard 积与 Manin 白积相等的条件，等价于二次部分控制 2-簇及 Dong 性质，并区分白积与其二次近似。
The operad of anti-pre-Lie algebras and its Lie elements
我们开启了针对反预李代数的系统 operadic 研究方法。研究了控制反预李代数的 operad、其与 Novikov 代数 operad 的关系及其循环结构。证明反预李代数的 Lie 括号满足五次标准恒等式，还证明反预李 operad 相对于该恒等式具有二次极大性。随后表明反预李 operad 及其 Koszul 对偶均非 Koszul，并完整描述了后者。最后，将 Koszul 对偶 operad 中的 Lie 元素与 metabelian Lie 代数的 operad 对应起来。
Compatible structures of operads by polarization, their Koszul duality and Manin products
该研究将不变量理论的极化推广到operad层面，统一了线性、匹配、全相容三类operad结构，揭示了其Koszul对偶性质与Manin乘积构造，且部分有限生成二元二次operad的Koszul性在相容构造中保持。
We consider the bar complex of a monomial non-unital associative algebra $A=k \langle X \rangle / (w_1,...,w_t)$. It splits as a direct sum of complexes $B_w$, defined for any fixed monomial $w=x_1...x_n \in A$. We give a simple argument, showing that the homology of this subcomplex is at most one-dimensional, and describe the place where the nontrivial homology appears. It has a very simple expression in terms of the length of the generalized Dyck path associated to a given monomial in $w \in A$. The operadic analogue of the question about dichotomy in homology is considered. It is shown that…
We present a study of quadratic operads for n-ary algebras and their dual for n odd. We will focus on the ternary case (i.e n=3). The aim is to underline the problem of computing the dual operad and the fact that this last is in general defined in the graded differential operad framework. We prove that the operad associated to 3-ary partially associative algebra is not Koszul. Recall that, if n is even, the operad of n-ary partially associative algebras is Koszul.
The Veronese square of the dendriform operad
本文研究了树形运算的Veronese平方，通过选择五个生成操作并利用代数方法确定了33个线性无关的二次关系。
In this work we provide a definition of a coloured operad as a monoid in some monoidal category, and develop the machinery of Gröbner bases for coloured operads. Among the examples for which we show the existance of a quadratic Gröbner basis we consider the seminal Lie-Rinehart operad whose algebras include pairs (functions, vector fields).
The cup product in the cohomology of algebras over quadratic operads has been studied in the general setting of Koszul duality for operads. We study the cup product on the cohomology of n-ary totally associative algebras with an operation of even (homological) degree. This cup product endows the cohomology with the structure of an n-ary partially associative algebra with an operation of even or odd degree depending on the parity of n. In the cases n = 3 and n = 4, we provide an explicit definition of this cup product and prove its basis properties.
A Lie-admissible algebra gives by anticommutativity a Lie algebra. In this work we study remarkable classes of Lie-admissible algebras such as Vinberg, PreLie algebras. We compute the corresponding binary quadratic operads and study their Koszul duality. Considering Lie algebras as Lie-admissible algebras we can define for each Lie algebra a cohomology with values in a Lie-admissible module. This permits to study some deformations of Lie algebras in the category of Lie-admissible algebras. Lastly we study the tensor product between these operads and their dual operads. As application we constr…
On the differential Novikov algebras
本文研究带有导子的 Novikov 代数及其对偶 operad，发现该对偶 operad 与双交换代数存在联系，研究动机源于 Novikov operad 的白与黑 Manin 乘积。
Coherent unit actions on a binary, quadratic operad were introduced by Loday and were shown by him to give Hopf algebra structures on the free algebras when the operad is also regular with a splitting of associativity. Working with such operads, we characterize coherent unit actions in terms of linear equations of the generators of the operads. We then use these equations to give all possible operad relations that allow such coherent unit actions. We further show that coherent unit actions are preserved under taking products and thus yield Hopf algebras on the free object of the product operad…
Universal Coacting Hopf Algebra of a finite-dimensional Algebra over an Operad
本文将泛共作用 Hopf 代数的构造推广到对称 operad 上的任意有限维代数，证明其范畴富集性并刻画自同构与分次，覆盖现有代数情形且扩展至分次对称 operad 上的代数。
We investigate the action of Schur algebra on the Lie algebras of derivations of free Lie algebras and operad structures constructed from it. We also show that the Lie algebra of derivations is generated by quadratic derivations together with the action of the Schur operad. Applications to certain subgroups of the automorphism group of a finitely generated free group are given as well.
Extended (tri)dendriform algebras, pre-Lie algebras and post-Lie algebras as companion structures of extended Rota-Baxter algebras
本文引入扩展（三）枝形代数、pre-Lie代数与post-Lie代数，建立其结构关系，证明它们可由扩展Rota-Baxter算子导出，并通过带括号的字构造自由扩展Rota-Baxter代数。
This paper provides a conceptual study of the twisting procedure, which amounts to create functorially new differential graded Lie algebras, associative algebras or operads (as well as their homotopy versions) from a Maurer--Cartan element. On the way, we settle the integration theory of complete pre-Lie algebras in order to describe this twisting procedure in terms of gauge group action. We give a criterion on quadratic operads for the existence of a meaningful twisting procedure of their associated categories of (homotopy) algebras. We also give a new presentation of the twisting procedure f…
We discuss algorithmic approach to growth of the codimension sequences of varieties of multilinear algebras, or, equivalently, the sequences of the component dimensions of algebraic operads. The (exponentional) generating functions of such sequences are called codimension series of varieties, or generating series of operads. We show that in general there does not exist an algorithm to decide whether the growth exponent of a codimension sequence of a variety defined by given finite sets of operations and identities is equal to a given rational number. In particular, we solve negatively a recent…
Koszul duality, minimal model and $L_\infty$-structure for differential algebras with weight
本文继续Loday的研究，为非零权情况下的微分代数operad建立了Koszul对偶，确定了极小模型与Koszul对偶同伦cooperad，并推导出带权同伦微分代数概念及其形变复形的$L_\infty$-代数结构。
An infinitesimal deformation of the post-Lie and post-Hopf algebra correspondence
该研究证明后李与后霍普夫代数的伴随关系与无穷小结构相容，推广了Cartier–Milnor–Moore定理，完成了相关结构的分类并证明了无穷小后李代数operad的Koszul性。
Maurer-Cartan methods in deformation theory: the twisting procedure
本专著概述了形变理论中的Maurer-Cartan方法，核心在于系统阐述twisting procedure，即从Maurer-Cartan元素函子地构造新的微分分次Lie代数、结合代数或operad，并给出了二次operad存在该过程的准则，同时结合图同调等示例展示其应用。
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