Tao Zhang, Ling Zhang, Ruyi Xie
The extending structures and unified products for Malcev algebras are developed. Some special cases of unified products such as crossed products and matched pair of Malcev algebras are studied. It is proved that the extending structures can be classified by some non-abelian cohomology theory. One dimensional flag extending structures of Malcev algebras are also investigated.
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Lie bialgebras were introduced by Drinfeld in studying the solutions to the classical Yang-Baxter equation. The definition of a bialgebra in the sense of Drinfeld (D-bialgebra), related with any variety of algebras, was given by Zhelyabin. In this work, we consider Malcev bialgebras. We describe all structures of a Malcev bialgebra on a simple non-Lie Malcev algebra.
In a Hom-Malcev algebra an identity, equivalent to the Hom-Malcev identity, is found.
We defined (n+1)-ary derivations of n-ary algebras. We described 3-derivations of simple Malcev algebras and 4-ary derivations of simple ternary Malcev algebra M_8.
The main feature of Hom-algebras is that the identities defining the structures are twisted by linear maps. The purpose of this paper is to introduce and study a Hom-type generalization of pre-Malcev algebras and M-dendriform algebras, called Hom-pre-Malcev algebras and Hom-M-dendriform algebras. We also introduce the notion of $\mathcal{O}$-operators of Hom-Malcev and Hom-pre-Malcev algebras and show the connections between Hom-Malcev, Hom-pre-Malcev and Hom-M-dendriform algebras using $\mathcal{O}$-operators. Hom-pre-Malcev algebras and Hom-M-dendriform algebras generalize Hom-pre-Lie algebr…
On flat pseudo-Euclidean solvable Malcev algebras
本文为伪欧几里得Malcev代数引入曲率算子，定义平坦性，发展平坦双扩张构造，证明平坦洛伦兹幂零Malcev代数必为Lie代数且平坦。
A Hom-Lie-Yamaguti algebra, whose ternary operation expresses through its binary one in a specific way, is a multiplicative Hom-Malcev algebra. Any multiplicative Hom-Malcev algebra over a field of characteristic zero has a natural Hom-Lie-Yamaguti structure.
Every multiplicative Hom-Malcev algebra has a natural multiplicative Hom-Lie triple system structure. Moreover, there is a natural Hom-Bol algebra structure on every multiplicative Hom-Malcev algebra and on every multiplicative right (or left) Hom-alternative algebra.
We find a basis of the free Malcev algebra on three free generators over a field of characteristic zero. The specialty and semiprimity of this algebra are proved. In addition, we prove the decomposability of this algebra into subdirect sum of the free Lie algebra rank three and the free algebra of rank three of variety of Malcev algebras generated by a simple seven-dimensional Malcev algebra.
The aim of this paper is to extend Gerstenhaber formal deformations of algebras to the case of Hom-Alternative and Hom-Malcev algebras. We construct deformation cohomology groups in low dimensions. Using a composition construction, we give a procedure to provide deformations of alternative algebras (resp. Malcev algebras) into Hom-alternative algebras (resp. Hom-Malcev algebras). Then it is used to supply examples for which we compute some cohomology invariants.
The purpose of this paper is to introduce and study BiHom-alternative algebras and BiHom-Malcev algebras. It is shown that BiHom-alternative algebras are BiHom-Malcev admissible and BiHom-Jordan admissible. Moreover, BiHom-type generalizations of some well known identities in alternative algebras, including the Moufang identities, are obtained.
We introduce two classes of nonassociative algebras and define the building blocks in the context of the new nonassociative algebras.
On the Lie algebra of a Malcev algebra
本文通过引入并利用唯一对应于 Malcev 代数的李代数，推导了其理想的基本性质，并证明了根空间分解的精确性质，从而发展了 Malcev 代数的结构理论。
Lie algebras endowed with an action by automorphisms of any of the symmetric groups S3 or S4 are considered, and their decomposition into a direct sum of irreducible modules for the given action is studied. In case of S3-symmetry, the Lie algebras are coordinatized by some nonassociative systems, which are termed generalized Malcev algebras, as they extend the classical Malcev algebras. These systems are endowed with a binary and a ternary products, and include both the Malcev algebras and the Jordan triple systems.
In this paper we give a construction of the universal enveloping algebra of a Malcev algebra in categories of group algebra comodules with a symmetry given by a bicharacter of the group. A particular example of such categories is the category of super vector spaces.
We describe degenerations of four-dimensional binary Lie algebras, and five- and six-dimensional nilpotent Malcev algebras over \mathbb{C}. In particular, we describe all irreducible components of these varieties.
We introduce and investigate the algebras of steadily growing length, that is the class of algebras, where the length is bounded by a linear function of the dimension. In particular we show that Malcev algebras belong to this class and establish the exact upper bound for its length.
In this note we describe the structure of finite dimensional Malcev algebras over a field of real numbers $\mathbb{R}$, which are nilpotent module its Lie center. It is proved that the corresponding analytic global Moufang loops are nilpotent module their nucleus.
In this paper we define pre-Malcev algebras and alternative quadri-algebras and prove that they generalize pre-Lie algebras and quadri-algebras respectively to the alternative setting. Constructions in terms bimodules, splitting of operations, and Rota-Baxter operators are discussed.
We introduce the class of split Malcev-Poisson-Jordan algebras as the natural extension of the one of split Malcev Poisson algebras, and therefore split (non-commutative) Poisson algebras. We show that a split Malcev-Poisson-Jordan algebra $P$ can be written as a direct sum $P = \oplus_{j \in J}I_j$ with any $I_j$ a non-zero ideal of $P$ in such a way that satisfies $[I_{j_1},I_{j_2}] = I_{j_1} \circ I_{j_2} = 0$ for $j_1 \neq j_2.$ Under certain conditions, it is shown that the above decomposition of $P$ is by means of the family of its simple ideals.
In this note we introduce the concept of an almost left automorphic Moufang loop and study the properties of tangent algebras of smooth loops of this class.
Kupershmidt-Nijenhuis structures on pre-Malcev algebras
本文研究了预马尔可夫代数上的库普辛斯基-尼金斯结构，构造了新的复预马尔可夫代数，并利用计算理想理论方法刻画其几何结构。
Bialgebras, Manin triples of Malcev-Poisson algebras and post-Malcev-Poisson algebras
本文引入 Malcev-Poisson 双代数与 post-Malcev-Poisson 代数，建立匹配对、Manin 三元组与双代数的等价，并揭示其作为加权相对 Rota-Baxter 算子底层结构的作用。
On a problem by Nathan Jacobson for Malcev algebras
本文解决了 Malcev 代数中类似于 Jacobson 对交错代数提出的问题，证明了包含三维简单李代数的 Malcev 代数的坐标化定理，并描述了去掉条件后的结构。
The simple 7-dimensional Malcev algebra $M$ is isomorphic to the irreducible $\mathfrak{sl}(2,\mathbb{C})$-module V(6) with binary product $[x,y] = α(x \wedge y)$ defined by the $\mathfrak{sl}(2,\mathbb{C})$-module morphism $α\colon Λ^2 V(6) \to V(6)$. Combining this with the ternary product $(x,y,z) = β(x \wedge y) \cdot z$ defined by the $\mathfrak{sl}(2,\mathbb{C})$-module morphism $β\colon Λ^2 V(6) \to V(2) \approx \s$ gives $M$ the structure of a generalized Lie triple system, or Lie-Yamaguti algebra. We use computer algebra to determine the polynomial identities of low degree satisfied b…
Let $A$ be a finite dimensional associative algebra over a perfect field and let $R$ be the radical of $A$. We show that for every one-sided ideal $I$ of $A$ there exists a semisimple subalgebra $S$ of $A$ such that $I=I_{S}\oplus I_{R}$ where $I_{S}=I\cap S$. and $I_{R}=I\cap R$.
Malcev classification for the variety of left-symmetric algebras
本文研究了左对称代数变种中的三个子变种，揭示其与交替、asso对称和Zinbiel代数的关系，并推导了相关运算在度数至4时的恒等式。
The purpose of this paper is to study the $\mathcal{O}$-operators on Malcev algebras and discuss the solutions of Malcev Yang-Baxter equation by $\mathcal{O}$-operators. Furthermore we introduce the notion of weighted $\mathcal{O}$-operators on Malcev algebras, which can be characterized by graphs of the semi-direct product Malcev algebra. Then we introduce a new algebraic structure called post-Malcev algebras. Therefore, post-Malcev algebras can be viewed as the underlying algebraic structures of weighted $\mathcal{O}$-operators on Malcev algebras. A post-Malcev algebra also gives rise to a n…
The main result is to prove that if a Malcev algebra $A$ is \textit{right nilpotent} of degree $n$, then $A$ is \textit{strongly nilpotent} of degree less or equals to $4n^2-2n+1$.
Moens proved that a finite-dimensional Lie algebra over field of characteristic zero is nilpotent if and only if it has an invertible Leibniz-derivation. In this article we prove the analogous results for finite-dimensional Malcev, Jordan, (-1,1)-, quasiassociative, quasialternative, right alternative and Malcev-admissible noncommutative Jordan algebras over the field of characteristic zero. Also, we describe all Leibniz-derivations of semisimple Jordan, right alternative and Malcev algebras.
Representation and cohomology of embedding tensors on Malcev algebras
本文引入Malcev代数上嵌入张量的概念，研究其表示与上同调理论，并探讨在形变与阿贝尔扩张中的应用。
Let K be a skew field and (G,<) an ordered group. We show that the skew field generated by the group ring K[G] inside the Malcev-Neumann series ring K((G;<)) contains noncommutative free group algebras.
Kronecker factorization theorems for the exceptional Malcev algebra
本文证明包含 7 维简单非 Lie Malcev 代数 $\mathbb{M}$ 且满足非退化条件的 Malcev 代数及超代数分别同构于 $\mathbb{M}$ 与交换结合代数或超交换结合超代数的张量积。
The purpose of this paper is to provide and study a Hom-type generalization of Jordan-Malcev-Poisson algebras, called Hom-Jordan-Malcev-Poisson algebras. We show that they are closed under twisting by suitable self-maps and give a characterization of admissible Hom-Jordan-Malcev-Poisson algebras. In addition, we introduce the notion of pseudo-Euclidian Hom-Jordan-Malcev-Poisson algebras and describe its $T^*$-extension. Finally, we generalize the notion of Lie-Jordan-Poisson triple system to the Hom setting and establish its relationships with Hom-Jordan-Malcev-Poisson algebras.
The main feature of Hom-algebras is that the identities defining the structures are twisted by linear maps. The purpose of this paper is to introduce and study a Hom-type generalization of pre-Malcev algebras, called Hom-pre-Malcev algebras. We also introduce the notion of Kupershmidt operators of Hom-Malcev and Hom-pre-Malcev algebras and show the connections between Hom-Malcev and Hom-pre-Malcev algebras using Kupershmidt operators. Hom-pre-Malcev algebras generalize Hom-pre-Lie algebras to the Hom-alternative setting and fit into a bigger framework with a close relationship with Hom-pre-alt…
The present paper is devoted to provide conditions for the Levi--Malcev theorem to hold or not to hold (i.e. for two Levi subalgebras to be or not conjugate by an inner automorphism) in the context of finite-dimensional Leibniz algebras over a field of characteristic zero. Particularly, in the case of the field $\mathbb{C}$ of complex numbers, we consider all possible cases in which Levi subalgebras are conjugate and not conjugate.
We study the universal enveloping algebras of the one-parameter family of solvable 5-dimensional non-Lie Malcev algebras. We explicitly determine the universal nonassociative enveloping algebras (in the sense of Perez-Izquierdo and Shestakov) and the centers of the universal enveloping algebras. We also determine the universal alternative enveloping algebras.
The main goal of this work is to introduce the notion of Hom-M-dendriform algebras which are the dendriform version of Hom-Malcev algebras. In fact they are the algebraic structures behind the $\mathcal{O}$-operator of Hom-pre-Malcev algebras. They also fit into a bigger framework as Hom-Malcev algebraic analogues of Hom-L-dendriform algebras. Furthermore, we show a connections between Hom-M-Dendriform algebras and Hom-alternative quadri-algebras.
We first discuss the construction by Perez-Izquierdo and Shestakov of universal nonassociative enveloping algebras of Malcev algebras. We then describe recent results on explicit structure constants for the universal enveloping algebras (both nonassociative and alternative) of the 4-dimensional solvable Malcev algebra and the 5-dimensional nilpotent Malcev algebra. We include a proof (due to Shestakov) that the universal alternative enveloping algebra of the real 7-dimensional simple Malcev algebra is isomorphic to the 8-dimensional division algebra of real octonions. We conclude with some bri…
Skew-symmetric solutions of the classical Yang-Baxter equation and $\mathcal{O}$-operators of Malcev algebras
研究Malcev代数中经典Yang-Baxter方程斜对称解与$\mathcal{O}$-算子的联系，证明二者在余伴随表示下可相互转化，并推广至pre-Malcev代数。
Perez-Izquierdo and Shestakov recently extended the PBW theorem to Malcev algebras. It follows from their construction that for any Malcev algebra $M$ over a field of characteristic $\ne 2, 3$ there is a representation of the universal nonassociative enveloping algebra $U(M)$ by linear operators on the polynomial algebra $P(M)$. For the nilpotent non-Lie Malcev algebra $\mathbb{M}$ of dimension 5, we use this representation to determine explicit structure constants for $U(\mathbb{M})$; from this it follows that $U(\mathbb{M})$ is not power-associative. We obtain a finite set of generators for …
We apply Kolesnikov's algorithm to obtain a variety of nonassociative algebras defined by right anticommutativity and a `noncommutative' version of the Malcev identity. We use computational linear algebra to verify that these identities are equivalent to the identities of degree <= 4 satisfied by the dicommutator in every alternative dialgebra. We extend these computations to show that any special identity for Malcev dialgebras must have degree at least 7. Finally, we introduce a trilinear operation which makes any Malcev dialgebra into a Leibniz triple system.
We prove that every clone of operations on a finite set A, if it contains a Malcev operation, is finitely related -- i.e., identical with the clone of all operations respecting R for some finitary relation R over A. It follows that for a fixed finite set A, the set of all such Malcev clones is countable. This completes the solution of a problem that was first formulated in 1980, or earlier: how many Malcev clones can finite sets support? More generally, we prove that every finite algebra with few subpowers has a finitely related clone of term operations. Hence modulo term equivalence and a ren…
Explicit Baker--Campbell--Hausdorff Radii in Special Banach--Malcev Algebras of Shifts
本文在特殊移位 Banach–Malcev 代数中建立了 BCH 级数的显式收敛半径，利用 Catalan 数控制得到精确常数 1/(4K)，并针对多种移位代数显式计算了连续性常数 B。
The purpose of this paper is to study Sabinin algebras of Hom-type. It is shown that Lie, Malcev, Bol and other algebras of Hom-type are naturally Sabinin algebras of Hom-type. To this end, we provide a general key construction that establish a relationship between identities of some class of Hom-algebras and ordinary algebras. Moreover, we discuss a new concept of Hom-bialgebra, in relation with universal enveloping Hom-algebras. A study based on primitive elements is provided.
Spectral Theory and Almost Periodic Structures in Hom--Lie Banach Algebras
本文提出Hom-李Banach代数的谱理论框架，揭示了扭曲导数的谱分解及几乎周期子空间的结构特性，并通过具体例子展示了其在算子代数中的应用。
Classification of Lie algebras constructed from $\mathfrak{gl}_{m|n}$ via Derived Bracket
本文通过奇元素 $B$ 生成的派生括号，对一般线性李超代数 $\mathfrak{gl}_{m|n}$ 构造的一族李代数进行了完全的结构和同构分类，证明了同构类型完全由秩 $r$ 和维数对决定，并给出了 Levi-Malcev 分解。
In this paper we study twisted algebras of multiplier Hopf ($^*$-)algebras which generalize all kinds of smash products such as generalized smash products, twisted smash products, diagonal crossed products, L-R-smash products, two-sided crossed products and two-sided smash products for the ordinary Hopf algebras appeared in [P-O].
We reformulate the definition of a zero product determined algebra in terms of tensor products and obtain necessary and sufficient conditions for an algebra to be zero product determined. These conditions allow us to prove that the direct sum \bigoplus_{i \in I} A_i of algebras for any index set I is zero product determined if and only if each of the component algebras A_i is zero product determined. As an application, every parabolic subalgebra of a finite-dimensional reductive Lie algebra, over an algebraically-closed field of characteristic zero, is zero product determined. In particular, e…
We prove that the Yoneda Ext-algebra of a Takeuchi smash product is the graded Takeuchi smash product of the Yoneda Ext-algebras of the two algebras or modules involved. As an application, we prove that graded Takeuchi smash products preserve Artin-Schelter regularity, and describe the Nakayama automorphism of the product.
We prove that assosymmetric algebras under Jordan product are Lie triple. A Lie triple algebra is called special if it is isomorphic to a subalgebra of some plus-assosymmetric algebra. We establish that Glennie identitiy is valid for special Lie triple algebras, but not for all Lie triple algebras.
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.
We discuss the Kummer subspaces of tensor products of cyclic algebras, focusing mainly on the case of cyclic algebras of degree 3. We present a family of maximal spaces in the general case, classify all the monomial spaces in the case of tensor products of cyclic algebras of degree 3 using graph theory, and provide an upper bound for the dimension in the generic tensor product of cyclic algebras of degree 3.
A five term sequence for the low degree cohomology of a smash product of (cocommutative) Hopf algebras is obtained, generalizing that of Tahara for a semi-direct product of groups.
We study BiHom-Novikov-Poisson algebras, which are twisted generalizations of Novikov-Poisson algebras and Hom-Novikov-Poisson algebras, and find that BiHom-Novikov-Poisson algebras are closed under tensor products and several kinds of perturbations. Necessary and sufficient conditions are given under which BiHom-Novikov-Poisson algebras give rise to BiHom-Poisson algebras.
The question of whether a split tensor product of quaternion algebras with involution over a field of characteristic two can be expressed as a tensor product of split quaternion algebras with involution, is shown to have an affirmative answer.
The extending structures problem for Zinbiel 2-algebras is studied. We introduce the concept of unified products for Zinbiel 2-algebras. Some special cases of unified products such as crossed product and matched pair of Zinbiel 2-algebras are investigated. It is proved that the extending problem for Zinbiel 2-algebras can be classified by some non-abelian cohomology theory.
We discuss finiteness/infiniteness of $τ$-tilting modules over tensor products of two symmetric algebras. As an application, we discuss that over block algebras of direct products of finite groups.
The mixed verbal wreath product in representations of Lie algebras is a construction parallel to the verbal wreath product of Lie algebras introduced by A. L. Shmelkin. It is shown that the analog of the group theorem on embedding in the verbal wreath product also holds for representations of Lie algebras.
Skew category algebras, twisted tensor product algebras and induction of precosheaves
本文证明斜范畴代数可嵌入扭曲张量积代数，并将Puig和Turull的概念从群代数推广到范畴代数，研究其在斜范畴代数下的性质。
We establish bounds for the representation dimension of skew group algebras and wreath products. Using this, we obtain bounds for the representation dimension of a block of a Hecke algebra of type A, in terms of the weight of the block. This includes certain blocks of group algebras of symmetric groups.
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