E. Remm
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.
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We present coalgebras of some classes of nonassociative algebras whose associator satisfies invariance conditions given by the action of the 3-order symmetric group. Amongst these algebras we find the well-known Vinberg algebras, the Pre-Lie algebras, the Lie-admissible algebras and the 3-power associative algebras. We study properties between these coalgebras and algebras which generalize the associative case.
In this paper, we introduce the definition of free product of operads, following the definition of a free product of algebras. There is a given method of finding the basis and dimension of the free product of operads. By anti-commutative operation, Lie-admissible algebra gives Lie algebra. We proved that Lie-admissible operad isomorphic to the free product of Lie and commutative(non-associative) operads.
On the free metabelian Novikov and metabelian Lie-admissible algebras
本文构造了自由亚交换Novikov代数与自由亚交换Lie-容许代数的线性基，并刻画了二者的对称多项式空间。
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.
M. Boucetta introduced the notion of pseudo-Riemannian Lie algebra in [2] when he studied the line Poisson structure on the dual of a Lie algebra. In this paper, we redefine pseudo-Riemannian Lie algebra, which, in essence, is a class of new Lie admissible algebras and prove that all pseudo-Riemannian Lie algebras are solvable. Using our main result and method, we prove some of M. Boucetta's results in [2, 3] in a simple and new way, then we give an explicit construction of Riemann-Lie algebras and a classification of pseudo-Riemannian Lie algebras of dimension 2 and 3.
In this article, we mainly study the products of commutator ideals of Lie-admissible algebras such as Novikov algebras, bicommutative algebras, and assosymmetric algebras. More precisely, we first study the properties of the lower central chains for Novikov algebras and bicommutative algebras. Then we show that for every Lie nilpotent Novikov algebra or Lie nilpotent bicommutative algebra $\mathcal{A}$, the ideal of $\mathcal{A}$ generated by the set $\{ab - ba\mid a, b\in \mathcal{A}\}$ is nilpotent. Finally, we study properties of the lower central chains for assosymmetric algebras, study th…
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…
The aim of this paper is to generalize the concept of Lie-admissible coalgebra introduced by Goze and Remm to Hom-coalgebras and to introduce Hom-Hopf algebras with some properties. These structures are based on the Hom-algebra structures introduced by the authors.
The purpose of this paper is to study Hom-Lie superalgebras, that is a superspace with a bracket for which the superJacobi identity is twisted by a homomorphism. This class is a particular case of $Γ$-graded quasi-Lie algebras introduced by Larsson and Silvestrov. In this paper, we characterize Hom-Lie admissible superalgebras and provide a construction theorem from which we derive a one parameter family of Hom-Lie superalgebras deforming the orthosymplectic Lie superalgebra. Also, we prove a $\mathbb{Z}_2$-graded version of a Hartwig-Larsson-Silvestrov Theorem which leads us to a construction…
We study Hom-type analogs of Rota-Baxter and dendriform algebras, called Rota-Baxter $G$-Hom-associative algebras and Hom-dendriform algebras. Several construction results are proved. Free algebras for these objects are explicitly constructed. Various functors between these categories, as well as an adjunction between the categories of Rota-Baxter Hom-associative algebras and of Hom-(tri)dendriform algebras, are constructed.
Some results on Hom-Novikov superalgebras and Hom-anti-pre-Novikov superalgebras
本文研究Hom-Novikov超代数的结构等性质，引入Hom-anti-pre-Novikov超代数，建立相关等价性、算子刻画及泊松型结构联系，明确其与可逆反超-𝒪-算子的基本关系。
We give an explicit description of commutative post-Lie algebra structures on some classes of nilpotent Lie algebras. For non-metabelian filiform nilpotent Lie algebras and Lie algebras of strictly upper-triangular matrices we show that all CPA-structures are associative and induce an associated Poisson-admissible algebra.
Matched pairs and double construction bialgebras of (transposed) Poisson 3-Lie algebras
本文利用匹配对研究 Poisson 3-李代数与转置 Poisson 3-李代数的双构造双代数，证明相关等价关系，并引入容许转置 Poisson 3-李代数以建立其双构造理论。
In this paper, we study admissible $ω$-left-symmetric algebraic structures on $ω$-Lie algebras over the complex numbers field $\mathbb C$. Based on the classification of $ω$-Lie algebras, we prove that any perfect $ω$-Lie algebra can't be the $ω$-Lie algebra defined by the commutator of an $ω$-left-symmetric algebra.
A Hom-algebra structure is a multiplication on a vector space where the structure is twisted by a homomorphism. The structure of Hom-Lie algebra was introduced by Hartwig, Larsson and Silvestrov and extended by Larsson and Silvestrov to quasi-hom Lie and quasi-Lie algebras. In this paper we introduce and study Hom-associative, Hom-Leibniz, and Hom-Lie admissible algebraic structures which generalize the well known associative, Leibniz and Lie admissible algebras. Also, we characterize the flexible Hom-algebras in this case. We also explain some connections between Hom-Lie algebras and Santilli…
Pre-Lie Structures for Semisimple Lie Algebras
本文研究了半单李代数与预李结构的可接受性，证明S3-结合代数是任何复数域上李代数的通用预李结构。
We present classes of nonassociative algebras whose associator satisfies invariance conditions given by the action of the 3 order symmetric group. Amongst these algebras we find the wellknown Pre Lie or Vinberg algebras and the Lie admissible algebras. We study the corresponding noncoassociative cogebras.
Averaging pre-Lie bialgebras
本文研究了平均预李代数的表示及其匹配对、马宁三元组和双代数理论，证明三者在特定条件下等价，并通过引入二次罗塔-巴克斯预李代数上的平均算子构造平均预李双代数，进一步展示相对罗塔-巴克斯算子生成可接受的经典Yang-Baxter方程的对称解。
The aim of this paper is to introduce the notion of BiHom-Lie superalgebras. This class of algebras is a generalization of both BiHom-Lie algebras and Hom-Lie superalgebras. In this article, we first present two ways to construct BiHom-Lie superalgebras from BiHom-associative superalgebras and Hom-Lie superalgebras by Yau's twist principle. Also, we explore some general classes of BiHom-Lie admissible superalgebras and describe all these classes via $G$-BiHom-associative superalgebras, where $G$ is a subgroup of the symmetric group $S_{3}$. Finally, we discuss the concept of $β^{k}$-derivation…
Given a symmetric Leibniz algebra $(\mathcal{L},.)$, the product is Lie-admissible and defines a Lie algebra bracket $[\;,\;]$ on $\mathcal{L}$. Let $G$ be the connected and simply-connected Lie group associated to $(\mathcal{L},[\;,\;])$. We endow $G$ with a Lie rack structure such that the right Leibniz algebra induced on $T_eG$ is exactly $(\mathcal{L},.)$. The obtained Lie rack is said to be associated to the symmetric Leibniz algebra $(\mathcal{L},.)$. We classify symmetric Leibniz algebras in dimension 3 and 4 and we determine all the associated Lie racks. Some of such Lie racks give ris…
We introduce the notions of pre-morphism and pre-derivation for arbitrary non-associative algebras over a commutative ring $k$ with identity. These notions are applied to the study of pre-Lie $k$-algebras and, more generally, Lie-admissible $k$-algebras. Associating with any algebra $(A,\cdot)$ its sub-adjacent anticommutative algebra $(A,[-,-])$ is a functor from the category of $k$-algebras with pre-morphisms to the category of anticommutative $k$-algebras. We describe the commutator of two ideals of a pre-Lie algebra, showing that the condition (Huq=Smith) holds for pre-Lie algebras. This a…
In this paper, we introduce the definition of generalized BiHom-Lie algebras and generalized BiHom-Lie admissible algebras in the category ${}_H{\mathcal M}$ of left modules for any quasitriangular Hopf algebra $(H, R) $. Also, we describe the BiHom-Lie ideal structures of the BiHom-associative algebras.
Keller admissible triples and Duflo theorem
本文研究Keller容许三元组，证明其可诱导一对关联微分分次代数的直和型Hochschild上同调的Gerstenhaber代数同构，并给出有限维李代数Duflo–Kontsevich定理的新具体证明。
Symplectic Leibniz algebras as a non-commutative version of symplectic Lie algebras
本文引入辛左/右 Leibniz 代数作为辛 Lie 代数的非交换推广，描述对称辛 Leibniz 代数，并利用双扩张与 $T^*$-扩张构造广泛例子。
This work provides a characterization of left and right Zinbiel algebras.Basic identities are established and discussed, showing that Zinbiel algebras are center-symmetric, and therefore Lie-admissible algebras. Their bimodules are given, and used to build a Zinbiel algebra structure on the direct sum of the underlying vector space and a finite-dimensional vector space. In addition, their matched pair is built, and related to the matched pair of their sub-adjacent Lie algebras. Besides, Zinbiel coalgebras are introduced, and linked to their underlying Lie coalgebras and coassociative coalgebra…
Lie admissible algebra structures, called center-symmetric algebras, are defined. Main properties and algebraic consequences are derived and discussed. Bimodules are given and used to build a center-symmetric algebra on the direct sum of underlying vector space and a finite dimensional vector space. Then, the matched pair of center-symmetric algebras is established and related to the matched pair of sub-adjacent Lie algebras. Besides, Manin triple of center-symmetric algebras is defined and linked with their associated matched pairs. Further, center-symmetric bialgebras of center-symmetric alg…
Actions of monoidal categories and representations of Cartan type Lie algebras
本文通过交叉同态建立幺半范畴作用，统一构造Lie-Rinehart代数及Leibniz对的弱表示与容许表示，并引入交叉同态上同调理论以研究其形变，同时为广义Witt代数等提供新表示。
This paper introduces the notion of Hom-Lie color algebra, which is a natural general- ization of Hom-Lie (super)algebras. Hom-Lie color algebras include also as special cases Lie (super) algebras and Lie color algebras. We study the homomorphism relation of Hom-Lie color algebras, and construct new algebras of such kind by a σ-twist. Hom-Lie color admissible algebras are also defined and investigated. They are finally classified via G-Hom-associative color algebras, where G is a subgroup of the symmetric group S_3.
The aim of this paper is to develop the theory of Hom-coalgebras and related structures. After reviewing some key constructions and examples of quasi-deformations of Lie algebras involving twisted derivations and giving rise to the class of quasi-Lie algebras incorporating Hom-Lie algebras, we describe the notion and some properties of Hom-algebras and provide examples. We introduce Hom-coalgebra structures, leading to the notions of Hom-bialgebra and Hom-Hopf algebras and prove some fundamental properties and give examples. Finally, we define the concept of Hom-Lie admissible Hom-coalgebra an…
The canonical dimension of modules for Iwasawa algebras
本文研究Iwasawa代数中模块的规范维度，证明了其无规范维度为一的模块，并给出了Iwasawa代数的Krull维度新上界，同时提出了一项关于有限长度表示的规范维度准则。
Cohomology theory of Nijenhuis Lie algebras and (generic) Nijenhuis Lie bialgebras
该文构建Nijenhuis李代数的上同调理论以刻画同时形变，拓展研究至一般Nijenhuis李双代数，引入匹配对与Manin三元组证明其等价性，并关联容许经典Yang-Baxter方程的反对称解。
We prove a structure theorem for finite-dimensional indefinite-signature metric 3-Lie algebras admitting a maximally isotropic centre. This algebraic condition indicates that all the negative-norm states in the associated Bagger-Lambert theory can be consistently decoupled from the physical Hilbert space. As an immediate application of the theorem, new examples beyond index 2 are constructed. The lagrangian for the Bagger-Lambert theory based on a general physically admissible 3-Lie algebra of this kind is obtained. Following an expansion around a suitable vacuum, the precise relationship betw…
Nijenhuis operators and mock-Lie bialgebras
本文通过推广匹配对、Manin三元组等概念，刻画了Nijenhuis mock-Lie双代数，并研究了其形式形变、阿贝尔扩张及与mock-Lie-Yang-Baxter方程的联系。
Recently, some concepts such as Hom-algebras, Hom-Lie algebras, Hom-Lie admissible algebras, Hom-coalgebras are studied and some of classical properties of algebras and some geometric objects are extended on them. In this paper by recall the concept of Hom-$ρ$-commutative algebras, we intend to develop some of the most classical results in Riemannian geometry such as metric, connection, torsion tensor, curvature tensor on it and also we discuss about differential operators and get some results of differential calculus using them. The notions of symplectic structures and Poisson structures are …
Left invariant affine structures in a Lie group $G$ are in one-to-one correspondence with left-symmetric algebras over its Lie algebra $\mathfrak g=T_eG$ (``over'' means that the commutator $[x,y]=xy-yx$ coincides with the Lie bracket; left-symmetric algebras can be defined as Lie-admissible algebras such that the multiplication by left defines a representation of the underlying Lie algebra). An affine structure (and the corresponding left symmetric algebra) is complete if $G$ is affinely equivalent to $\mathfrak g$. By the main result of this paper, a complete left symmetric algebra admits a …
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