Identities and derivations for Jacobian algebras

某些功能恒等式刻画三角代数上的双侧中心化子和双侧广义导数

Some functional identities characterizing two-sided centralizers and two-sided generalized derivations on triangular algebras

本文研究了三角代数上双侧中心化子和双侧广义导数的函数恒等式特性，并给出了相关结果。

Deformations of Lie Algebras using $σ$-derivations

In this article we develop an approach to deformations of the Witt and Virasoro algebras based on $σ$-derivations. We show that $σ$-twisted Jacobi type identity holds for generators of such deformations. For the $σ$-twisted generalization of Lie algebras modeled by this construction, we develop a theory of central extensions. We show that our approach can be used to construct new deformations of Lie algebras and their central extensions, which in particular include naturally the $q$-deformations of the Witt and Virasoro algebras associated to $q$-difference operators, providing also correspond…

Universal enveloping algebras and universal derivations of Poisson algebras

Let $k$ be an arbitrary field of characteristic $0$. It is shown that for any $n\geq 1$ the universal enveloping algebras of the Poisson symplectic algebra $P_n(k)$ and the Weyl algebra $A_n(k)$ are isomorphic and the canonical isomorphism between them easily leads to the Moyal product. A basis of the universal enveloping algebra $P^e$ of a free Poisson algebra $P=k\{x_1,...,x_n\}$ is constructed and proved that the left dependency of a finite number of elements of $P^e$ over $P^e$ is algorithmically recognizable. We prove that if two elements of a free Poisson algebra do not generate a free t…

Functional identities and zero Lie product determined Banach algebras

Three problems connecting functional identities to the recently introduced notion of a zero Lie product determined Banach algebra are discussed. The first one concerns commuting linear maps, the second one concerns derivations that preserve commutativity, and the third one concerns bijective commutativity preserving linear maps.

Multivariate Hasse-Schmidt Derivation on Exterior Algebras

The purpose of this short note is to consider multi-variate Hasse-Schmidt derivations on exterior algebras and to show how they easily provide remarkable identities, holding in the algebra of square matrices, which generalise the classical theorem of Cayley-Hamilton.

$\mathbb{Z}$-graded identities of the Lie algebras $U_1$

Let $K$ be an infinite field of characteristic different from two and let $U_1$ be the Lie algebra of the derivations of the algebra of Laurent polynomials $K[t,t^{-1}]$. The algebra $U_1$ admits a natural $\mathbb{Z}$-grading. We provide a basis for the graded identities of $U_1$ and prove that they do not admit any finite basis. Moreover, we provide a basis for the identities of certain graded Lie algebras with a grading such that every homogeneous component has dimension $\leq 1$, if a basis of the multilinear graded identities is known. As a consequence of this latter result we are able to…

Derivation double Lie algebras

We study classical R-matrices D for Lie algebras L such that D is also a derivation of L. This yields derivation double Lie algebras (L,D). The motivation comes from recent work on post-Lie algebra structures on pairs of Lie algebras arising in the study of nil-affine actions of Lie groups. We prove that there are no nontrivial simple derivation double Lie algebras, and study related Lie algebra identities for arbitrary Lie algebras.

微分恒等式与余维数的多项式增长

Differential identities and polynomial growth of the codimensions

本文研究李代数通过导子作用于结合代数形成的L-代数，在微分余维数序列多项式有界的条件下刻画其微分恒等式理想，并进一步刻画微分余特征重数被常数界定的情形。

Gelfand--Dorfman algebras, derived identities, and the Manin product of operads

Gelfand--Dorfman bialgebras (GD-algebras) are nonassociative systems with two bilinear operations satisfying a series of identities that express Hamiltonian property of an operator in the formal calculus of variations. The paper is devoted to the study of GD-algebras related with differential Poisson algebras. As a byproduct, we obtain a general description of identities that hold for operations $a\succ b = d(a)b$ and $a\prec b = ad(b)$ on a (non-associative) differential algebra with a derivation~$d$.

Wronskians as n-Lie multiplications

Filipov proved that Jacobian algebra is n-Lie. In our paper we consider algebras defined on associative commutative algebra U with derivation $\der$ by (k+1)-multiplication $V^{0,1,...,k}=\der^0\wedge\der^1\wedge...\wedge \der^k$ (Wronskian). We study whether they have (k+1)-Lie, k-left commutative and homotopical (k+1)-Lie structures.

$\mathbb{Z}$-graded identities of the Lie algebras $U_1$ in characteristic 2

Let $K$ be any field of characteristic two and let $U_1$ and $W_1$ be the Lie algebras of the derivations of the algebra of Laurent polynomials $K[t,t^{-1}]$ and of the polynomial ring $K[t]$, respectively. The algebras $U_1$ and $W_1$ are equipped with natural $\mathbb{Z}$-gradings. In this paper, we provide bases for the graded identities of $U_1$ and $W_1$, and we prove that they do not admit any finite basis.

Dhara-Rehman-Raza's identities on left ideals of prime rings

It is known that every nonzero Jordan ideal of $2$-torsion free semiprime rings contains a nonzero ideal. In this paper we show that also any square closed Lie ideal of a $2$-torsion free prime ring contains a nonzero ideal. This can be interpreted by saying that studying identities over one sided ideals is the "optimal" case to study identities. With this fact in mind, we generalize some results of Dhara, Rehman and Raza in [Lie ideals and action of generalized derivations in rings, Miskolc Mathematical Notes, \textbf{16} (2015), 769 -- 779] to the context of nonzero left ideals.

Identities of the left-symmetric Witt algebras

Let $P_n=k[x_1,x_2,\ldots,x_n]$ be the polynomial algebra over a field $k$ of characteristic zero in the variables $x_1,x_2,\ldots,x_n$ and $\mathscr{L}_n$ be the left-symmetric Witt algebra of all derivations of $P_n$. We describe all right operator identities of $\mathscr{L}_n$ and prove that the set of all algebras $\mathscr{L}_n$, where $n\geq 1$, generates the variety of all left-symmetric algebras. We also describe a class of general (not only right operator) identities for $\mathscr{L}_n$.

上三角矩阵的微分簇

Differential varieties of upper triangular matrices

本文研究上三角矩阵代数的微分恒等式，为微分指数为3的极小$L$-簇分类奠基，证明相关作用可约化，确定$UT_3$的微分恒等式与余维序列，且$k \geq 3$时的$UT_k$生成的$L$-簇必包含带对应作用的$UT_3$。

Quasi-hom-Lie Algebras, Central Extensions and 2-cocycle-like Identities

This paper begins by introducing the concept of a quasi-hom-Lie algebra which is a natural generalization of hom-Lie algebras introduced in a previous paper by the authors. Quasi-hom-Lie algebras include also as special cases (color) Lie algebras and superalgebras, and can be seen as deformations of these by homomorphisms, twisting the Jacobi identity and skew-symmetry. The natural realm for these quasi-hom-Lie algebras is as a generalization-deformation of the Witt algebra $\Witt$ of derivations on the Laurent polynomials $\C[t,t^{-1}]$. We also develop a theory of central extensions for qhl-…

矩阵代数的微分恒等式

Differential identities of matrix algebras

本文研究特征零域上矩阵代数在导子李代数作用下的微分恒等式，给出生成元、余维数及余特征标，并证明其生成簇具有几乎多项式增长。

结合代数上线性算子的代数恒等式的一种新分类

A new classification of algebraic identities for linear operators on associative algebras

本研究结合代数operad理论与计算交换代数方法，对结合代数上次数为2、重数为1或2的线性算子代数恒等式进行分类，得到多类已知及新的恒等式，且方法可推广至更一般情形。

自由非交换 Novikov 代数的基

Basis of the free noncommutative Novikov algebra

本文研究由带导子的结合代数推广得到的非交换 Novikov 代数，并为其自由代数构造了单项基。

单射极小模的简单n-李代数表示

Irreducible cuspidal modules of simple $n$-Lie algebras

研究简单n-李代数上Wronskians和Jacobian结构的不可约极小模的分类与保持不可约性证明。

微分排列代数中的恒等式

Identities in differential perm algebras

本文研究微分排列代数中由导出操作生成的多项式恒等式，探讨其代数结构及生成集，构建出自然的排列-威特型李代数和李贝恩兹代数。