Non-associative central extensions of null-filiform associative algebras

Abelian groups gradings on null-filiform and one-parametric filiform Leibniz algebras

We classify, up to equivalences, all abelian groups gradings on null-filiform and one-parametric filiform Leibniz algebras. Any grading on a null-filiform Leibniz algebra is toral but there are non-toral gradings on one-parametric filiform Leibniz algebras.

零丝结合代数上的转置 $δ$-Poisson 代数结构

Transposed $δ$-Poisson algebra structures on null-filiform associative algebras

本文研究了零丝结合代数上的转置 $δ$-Poisson 代数结构，给出了由多项式 $δ^3 - 3δ^2 + 2δ$ 的根刻画的完整分类，并构造了所有 $δ$-Poisson 代数结构，证明其为平凡结构。

On cohomology of filiform Lie superalgebras

Suppose the ground field $\mathbb{F}$ is an algebraically closed field of characteristic different from 2, 3. We determine the Betti numbers and make a decomposition of the associative superalgebra of the cohomology for the model filiform Lie superalgebra. We also describe the associative superalgebra structures of the (divided power) cohomology for some low-dimensional filiform Lie superalgebras.

Classification of solvable Leibniz algebras with null-filiform nilradical

In this paper we classify solvable Leibniz algebras whose nilradical is a null-filiform algebra. We extend the obtained classification to the case when the solvable Leibniz algebra is decomposed as a direct sum of its nilradical, which is a direct sum of null-filiform ideals, and a one-dimensional complementary subspace. Moreover, in this case we establish that these ideals are ideals of the algebra, as well.

Central extensions of filiform Zinbiel algebras

In this paper we describe central extensions (up to isomorphism) of all complex null-filiform and filiform Zinbiel algebras. It is proven that every non-split central extension of an $n$-dimensional null-filiform Zinbiel algebra is isomorphic to an $(n+1)$-dimensional null-filiform Zinbiel algebra. Moreover, we obtain all pairwise non isomorphic quasi-filiform Zinbiel algebras.

零幂层 Leibniz 代数上的多项式恒等式与多项式像

Polynomial identities and images of polynomials on null-filiform Leibniz algebras

研究零幂层 Leibniz 代数上的多项式恒等式与像，给出有限极小恒等式基，刻画多重齐次多项式像为子空间的条件，并验证多重线性情形的 L'vov-Kaplansky 类猜想。

几类可解 Leibniz 代数的双导子

Biderivations of some classes of solvable Leibniz algebras

本文研究 null-filiform、filiform 及相应可解 Leibniz 代数的反导子与双导子，刻画其一般形式，并证明相关可解情形的双导子均为内双导子。

Central extensions of null-filiform and naturally graded filiform non-Lie Leibniz algebras

In this paper we describe central extensions of some nilpotent Leibniz algebras. Namely, central extensions of the Leibniz algebra with maximal index of nilpotency are classified. Moreover, non-split central extensions of naturally graded filiform non-Lie Leibniz algebras are described up to isomorphism. It is shown that $k$-dimensional central extensions ($k\geq 5$) of these algebras are split.

有限维幂零结合代数的局部与2-局部自同构

Local and 2-local automorphisms of finite-dimensional nilpotent associative algebras

本文研究了$n$维null-filiform和filiform结合代数的自同构、局部与2-局部自同构，明确了其矩阵一般形式，发现局部自同构不全为自同构，并证明了所有2-局部自同构均为自同构。

最大长度拟丝状李代数上的转置泊松结构

Transposed Poisson structures on quasi-filiform Lie algebras of maximum length

该研究聚焦最大长度拟丝状李代数的$\frac{1}{2}$-导子，通过建立交换结合乘法，构造出以这类李代数为伴随代数的非平凡转置泊松代数，为相关代数结构研究提供新构造。

具有丝状幂零根基的可解李代数上的转置 Poisson 结构

Transposed Poisson structures on solvable Lie algebras with filiform nilradical

该研究利用具有线状幂零根基的可解李代数的 1/2-导子，建立交换结合乘法，从而构造出关联给定李代数的非平凡转置 Poisson 代数。

Degenerations of Poisson algebras

We construct a method to obtain the algebraic classification of Poisson algebras defined on a commutative associative algebra, and we apply it to obtain the classification of the $3$-dimensional Poisson algebras. In addition, we study the geometric classification, the graph of degenerations and the closures of the orbits of the variety of $3$-dimensional Poisson algebras. Finally, we also study the algebraic classification of the Poisson algebras defined on a commutative associative null-filiform or filiform algebra and, to enrich this classification, we study the degenerations between these p…

Infinitesimal deformations of null-filiform Leibniz superalgebras

In this paper we describe the infinitesimal deformations of null-filiform Leibniz superalgebras over a field of zero characteristic. It is known that up to isomorphism in each dimension there exist two such superalgebras $NF^{n,m}$. One of them is a Leibniz algebra (that is $m=0$) and the second one is a pure Leibniz superalgebra (that is $m\neq 0$) of maximum nilindex. We show that the closure of union of orbits of single-generated Leibniz algebras forms an irreducible component of the variety of Leibniz algebras. We prove that any single-generated Leibniz algebra is a linear integrable defor…

Automorphisms of some solvable Leibniz algebras

In this paper, we describe the automorphisms of solvable Leibniz algebras with null-filiform nilradical. Moreover we describe the automorphisms of solvable Leibniz algebras with naturally graded non-Lie filiform nilradicals, whose the dimension of complementary space is maximal.

三维反树形代数的分类

Classification of three-dimensional anti-dendriform algebras

本文分类了与结合性相关的反树形代数，重点处理零幂零结合代数及三维代数情形，并证明了含非零幂等元的结合代数不存在相容的反树形代数结构。

Local and 2-local automorphisms of some solvable Leibniz algebras

In this paper we prove that any local automorphism on the solvable Leibniz algebras with null-filiform and naturally graded non-Lie filiform nilradicals, whose dimension of complementary space is maximal is an automorphism. Furthermore, the same problem concerning 2-local automorphisms of such algebras is investigated and we obtain the analogously results for 2-local automorphisms.

Commutative Post-Lie algebra structures on nilpotent Lie algebras and Poisson algebras

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.

Classification of 4-dimensional nilpotent complex Leibniz algebras

The Leibniz algebras appeared as a generalization of the Lie algebras. In this work we deal with the classification of nilpotent complex Leibniz algebras of low dimensions. Namely, the classification of nilpotent complex Leibniz algebras dimensions less than 3 is extended to the dimension four. {\it AMS Subject Classifications}: 16D70, 17A30, 17A60, 17B30 {\it Key words:} Leibniz algebra, associative algebra, nilpotence, nulfiliform Leibniz algebra, filiform Leibniz algebra.

Zinbiel 超代数

Zinbiel superalgebras

本文将 Zinbiel 代数研究推广到超代数，证明其幂零性，并刻画复数域上零-filiform 单生成元超代数的唯一性及低维分类。

五维可解李代数的阿贝尔扩展

Abelian extensions of five-dimensional solvable Leibniz algebras

本文通过扩展中心扩展方法，对具有非平凡三维nilradical的五维可解李代数进行一维阿贝尔扩展的完整分类，并证明nilradical为null-filiform时存在唯一的可解阿贝尔扩展。

Associative superalgebras with homogeneous symmetric structures

A homogeneous symmetric structure on an associative superalgebra A is a non-degenerate, supersymmetric, homogeneous (i.e. even or odd) and associative bilinear form on A. In this paper, we show that any associative superalgebra with non null product can not admit simultaneously even-symmetric and odd-symmetric structure. We prove that all simple associative superalgebras admit either even-symmetric or odd-symmetric structure and we give explicitly, in every case, the homogeneous symmetric structures. We introduce some notions of generalized double extensions in order to give inductive descript…

On isomorphism conditions for algebra functors with applications to Leavitt path algebras

We introduce certain functors from the category of commutative rings (and related categories) to that of $\mathbb{Z}$-algebras (not necessarily associative or commutative). One of the motivating examples is the Leavitt path algebra functor $R\mapsto L_R(E)$ for a given graph $E$. Our goal is to find "descending" isomorphism results of the type: if $\mathfrak{F},\mathcal{G}$ are algebra functors and $K\subset K'$ a field extension, under what conditions an isomorphism $\mathfrak{F}(K')\cong \mathcal{G}(K')$ of $K'$-algebras implies the existence of an isomorphism $\mathfrak{F}(K)\cong\mathcal{G…

约旦代数、半复数与任意对称矩阵的Cholesky分解

Jordan algebras, hemiplex numbers, and the Cholesky decomposition of arbitrary symmetric matrices

研究任意对称矩阵Cholesky分解，通过类比正定矩阵与特定二次方程，利用半复数这种非结合代数实现分解，无需pivoting保持带状结构，对奇异矩阵可生成零空间参数化，为求解对称线性方程增添实用工具。