The non-symmetric operad pre-Lie is free

$n\text{-}Lie_d$ 算子及其 Koszul 对偶

$n\text{-}Lie_d$ Operad and its Koszul Dual

本文研究 $n\text{-}Lie_d$ 算子及其 Koszul 对偶 $n\text{-}Com_{-d+n-2}$，通过 Specht 模和奇图谱揭示两者联系。

共享一个李括号的 pre-Lie 乘积的组合学

Combinatorics of pre-Lie products sharing a Lie bracket

研究共享同一李括号的多个 pre-Lie 代数结构所对应的 operad，证明其具有类似 pre-Lie operad 的组合描述及诸多代数性质。

On Koszulity for operads of Conformal Field Theory

We study two closely related operads: the Gelfand-Dorfman operad GD and the Conformal Lie Operad CLie. The latter is the operad governing the Lie conformal algebra structure. We prove Koszulity of the Conformal Lie operad using the Groebner bases theory for operads and an operadic analogue of the Priddy criterion. An example of deformation of an operad coming from the Hom structures is considered. In particular we study possible deformations of the Associative operad from the point of view of the confluence property. Only one deformation, the operad which governs the identity $(α(ab))c=a(α(bc)…

Lie theory for Hopf operads

The present article takes advantage of the properties of algebras in the category of S-modules (twisted algebras) to investigate further the fine algebraic structure of Hopf operads. We prove that any Hopf operad P carries naturally the structure of twisted Hopf P-algebra. Many properties of classical Hopf algebraic structures are then shown to be encapsulated in the twisted Hopf algebraic structure of the corresponding Hopf operad. In particular, various classical theorems of Lie theory relating Lie polynomials to words (i.e. elements of the tensor algebra) are lifted to arbitrary Hopf operad…

Lie-admissible algebras and operads

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 Hopf algebra structures over free operads

The operad Lie can be constructed as the operad of primitives Prim As from the operad As of associative algebras. This is reflected by the theorems of Friedrichs, Poincare'-Birkhoff-Witt and Cartier-Milnor-Moore. We replace As by families of free operads P, which include the operad Mag freely generated by a noncommutative non-associative binary operation and the operad of Stasheff polytopes. We obtain Poincare'-Birkhoff-Witt type theorems and collect information about the operads Prim P, e.g. in terms of characteristic functions.

Pre-Lie algebras and the rooted trees operad

A Pre-Lie algebra is a vector space L endowed with a bilinear product * : L \times L to L satisfying the relation (x*y)*z-x*(y*z)= (x*z)*y-x*(z*y), for all x,y,z in L. We give an explicit combinatorial description in terms of rooted trees of the operad associated to this type of algebras and prove that it is a Koszul operad.

富集的预李操作元与自由性定理

Enriched pre-Lie operads and freeness theorems

本文研究了由Caleque和Willwacher定义的C富集预李操作元，用于构造作用于各种变形复形的操作元。Hopf合体之间的映射诱导出相应富集预李操作元之间的映射，并证明了域对余域的左右自由作用准则。

与非对称带乘法算畴相关的Frölicher-Nijenhuis括号和导出括号

Frölicher-Nijenhuis bracket and derived bracket associated to a nonsymmetric operad with multiplication

本文通过构造与非对称带乘法算畴相关的两个分次李代数，建立了Maurer-Cartan元素与Nijenhuis元素、Rota-Baxter元素的对应关系，并给出括号显式形式以研究Loday型代数上的相关算子。

Free pre-Lie family algebras

In this paper, we first define the pre-Lie family algebra associated to a dendriform family algebra in the case of a commutative semigroup. Then we construct a pre-Lie family algebra via typed decorated rooted trees, and we prove the freeness of this pre-Lie family algebra. We also construct pre-Lie family operad in terms of typed labeled rooted trees, and we obtain that the operad of pre-Lie family algebras is isomorphic to the operad of typed labeled rooted trees, which generalizes the result of F. Chapoton and M. Livernet. In the end, we construct Zinbiel and pre-Poisson family algebras and…

后继算子、格罗比纳-希尔绍夫基与自由L-双 dendriform 代数

Disuccessors, Gröbner-Shirshov bases and free L-dendriform algebras

本文证明结合、李、预李operad上的后继算子保持格罗比纳-希尔绍夫基，据此提出构造dend和pre-Lie operad基的新方法，显式构造自由L-双dendriform代数的基，为Madariaga问题提供高效解决方案并给出该代数组合结构的新见解。

On Schur Algebras and Derivations of Free Lie Algebras

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.

双结合（diassociative）与树状（dendriform）operad的单参数形变

One-parameter deformations of the diassociative and dendriform operads

本研究结合Livernet和Loday提出的结合operad极化构造与结合运算的树状分裂方法，借助二次operad的Koszul对偶，实现了非对称树状、双结合operad到对称operad范畴的单参数形变。

作为扩展Rota-Baxter代数伴随结构的扩展（三）枝形代数、pre-Lie代数和post-Lie代数

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。

Distributive laws between the Three Graces

By the Three Graces we refer, following J.-L. Loday, to the algebraic operads Ass, Com, and Lie, each generated by a single binary operation; algebras over these operads are respectively associative, commutative associative, and Lie. We classify all distributive laws (in the categorical sense of Beck) between these three operads. Some of our results depend on the computer algebra system Maple, especially its packages LinearAlgebra and Groebner.

代数算畴（Operad）的温和导论

A Gentle Introduction to Algebraic Operads

本文系统介绍代数算畴理论，证明结合、交换和李代数可分别作为算畴 As、Com、Lie 的表示范畴被统一恢复，并建立算畴与经典代数之间的范畴同构。

A class of nonassociative algebras including flexible and alternative algebras, operads and deformations

There exists two types of nonassociative algebras whose associator satisfies a symmetric relation associated with a 1-dimensional invariant vector space with respect to the natural action of the symmetric group on three elements. The first one corresponds to the Lie-admissible algebras that we studied in a previous paper. Here we are interested by the second one corresponding to the third power associative algebras.

与$n$-李双代数相关的Manin三元组

Manin triples associated to $n$-Lie bialgebras

该文围绕$n$-李双代数相关的Manin三元组展开研究，引入operad矩阵概念，通过其特殊情形导出局部上循环$n$-李双代数，还建立了$n$-李双代数偶对与$n$-李代数Manin三元组的一一对应。

受限李代数和受限李三系上的受限（相对）Rota-Baxter算子及相关结构

Restricted (Relative) Rota-Baxter operators on restricted Lie algebras and restricted Lie triple systems and related structures

本文在受限李代数和受限李三系上定义并研究受限Rota-Baxter算子，证明pre-Lie三系operad是Lie三系operad的split，并在正特征下证明Jacobson恒等式成立，进而引入受限pre-Lie三系。

Homotopy morphisms between convolution homotopy Lie algebras

In previous works by the authors, a bifunctor was associated to any operadic twisting morphism, taking a coalgebra over a cooperad and an algebra over an operad, and giving back the space of (graded) linear maps between them endowed with a homotopy Lie algebra structure. We build on this result by using a more general notion of $\infty$-morphism between (co)algebras over a (co)operad associated to a twisting morphism, and show that this bifunctor can be extended to take such $\infty$-morphisms in either one of its two slots. We also provide a counterexample proving that it cannot be coherently…

A Lie theoretic approach to the twisting procedure and Maurer-Cartan simplicial sets over arbitrary rings

The Deligne-Getzler-Hinich--$\infty$-groupoid or Maurer-Cartan simplicial set of an $L_\infty$-algebra plays an important role in deformation theory and many other areas of mathematics. Unfortunately, this construction only works over a field of characteristic $0$. The goal of this paper is to show that the notions of Maurer-Cartan equation and Maurer-Cartan simplicial set can be defined for a much larger number of operads than just the $L_\infty$-operad. More precisely, we show that the Koszul dual of every unital Hopf cooperad (a cooperad in the category of unital associative algebras) with …

A diagram connecting Symmetric Leibniz algebras with Lie-admissible algebras

In this paper we study symmetric Leibniz and related algebras, namely symmetric dialgebras and symmetric Perm-algebras. We also calculate their Koszul duals, if not known. This will give us Lie-admissible algebras and new types of algebras, which we call commutative and associative admissible algebras. Finally we prove that all operads describing the considered algebras are Koszul.

Free pre-Lie algebras are free as Lie algebras

We prove that free pre-Lie algebras, when considered as Lie algebras, are free. Working in the category of S-modules, we define a natural filtration on the space of generators. We also relate the symmetric group action on generators with the structure of the anticyclic PreLie operad.

The twisting procedure

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…

Novikov 代数的上同调理论及其应用

Cohomology theory of Novikov algebras and applications

本文通过 operad 方法刻画 Novikov 代数上同调，给出显式链复形，并证明其与底层 pre-Lie 代数上同调存在长正合序列，且第二上同调群分类无穷小形变与 Abel 扩张。

Pre-Lie algebras in positive characteristic

In prime characteristic we introduce the notion of restricted pre-Lie algebras. We prove in the pre-Lie context the analogue to Jacobson's theorem for restricted Lie algebras. In particular, we prove that any dendriform algebra over a field of positive characteristic is a restricted pre-Lie algebra. Thus we obtain that Rota-Baxter algebras and quasitriangular algebras are restricted pre-Lie algebras. Moreover, we prove that the free $Γ(\calligra{preLie})$-algebra is a restricted pre-Lie algebra, where $\calligra{preLie}$ denotes the pre-Lie operad. Finally, we define the notion of restricted e…

后李代数与后霍普夫代数对应关系的无穷小形变

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。