相关论文: On varieties of Lie algebras with infinite basis p…

We study locally finite varieties (=primitive classes) of linear algebras over finite fields. We do not assume that our algebras are associative or Lie. We are interested in the basic properties of finite algebras in these varieties such…

In recent years, the notion of characteristic polynomial of representations of Lie algebras has been widely studied. This paper provides more properties of these characteristic polynomials. For simple Lie algebras, we characterize the…

A new class of infinite dimensional simple Lie algebras over a field with characteristic 0 are constructed. These are examples of non-graded Lie algebras. The isomorphism classes of these Lie algebras are determined. The structure space of…

We characterize finite-dimensional Lie algebras over an arbitrary field of characteristic zero which admit a non-trivial (quasi-) triangular Lie bialgebra structure.

An algebra $L$ over a field $\Bbb F$, in which product is denoted by $[\,,\,]$, is said to be \textit{ Lie type algebra} if for all elements $a,b,c\in L$ there exist $\alpha, \beta\in \Bbb F$ such that $\alpha\neq 0$ and $[[a,b],c]=\alpha…

This is a survey on the finite basis problem for varieties of algebraic systems. Our exposition is in two directions: (i) We give numerous examples of varieties which are not finitely based. (ii) We give examples of important varieties with…

We consider some special type extensions of an arbitrary Lie algebra, which we call universal extensions. We show that these extensions are in one-to-one correspondence with finite dimensional associative commutative algebras. We also…

The Lie algebra of planar vector fields with coefficients from the field of rational functions over an algebraically closed field of characteristic zero is considered. We find all finite-dimensional Lie algebras that can be realized as…

We present an overview of characteristic identities for Lie algebras and superalgebras. We outline methods that employ these characteristic identities to deduce matrix elements of finite dimensional representations. To demonstrate the…

In this paper I consider locally finite Lie algebras of characteristic zero satisfying the condition that for every finite number of elements $x_{1}, x_{2},..., x_{k}$ of such an algebra $L$ there is finite-dimensional subalgebra $A$ which…

In this paper we focus on the structure of the variety of Lie algebras with a finite number of ideals and their graph representations using Hasse diagrams. The large number of necessary conditions on the algebraic structure of this type of…

Let X be an affine variety and L be a solvable Lie subalgebra of Lie(Aut(X)) generated by a finite collection of locally finite Lie subalgebras. The authors of [arXiv:2507.09679] wondered whether L is itself locally finite. Here we present…

We study a new class of infinite dimensional Lie algebras, which has important applications to the theory of integrable equations. The construction of these algebras is very similar to the one for automorphic functions and this motivates…

It is well known that a finite-dimensional Lie algebra over a field of characteristic zero is simple exactly when its derivation algebra is simple. In this paper we characterize those Lie algebras of arbitrary dimension over any field that…

We give an algorithm for constructing the algebraic hull of a given matrix Lie algebra in characteristic zero. It is based on an algorithm for finding integral linear dependencies of the roots of a polynomial, that is probably of…

In this article we describe varieties of Lie algebras via algebraic exponentiation, a concept introduced by Gray in his Ph.D. thesis. For $\mathbb{K}$ an infinite field of characteristic different from $2$, we prove that the variety of Lie…

The paper presents two new results concerning the varieties of Leibnitz algebras. In the case of prime characteristic p of the base field constructed example not nilpotent variety of Leibnitz algebras satisfying an Engel condition order p.…

We present a local and constructive differential geometric description of finite-dimensional solvable and transitive Lie algebras of vector fields. We show that it implies a Lie's conjecture for such Lie algebras. Also infinite-dimensional…

We consider associative algebras over a field. An algebra variety is said to be {\em Lie nilpotent} if it satisfies a polynomial identity of the kind $[x_1, x_2, ..., x_n] = 0$ where $[x_1,x_2] = x_1x_2 - x_2x_1$ and $[x_1, x_2, ..., x_n]$…

Let $A$ be an associative algebra over a field of characteristic $\neq 2$ that is generated by a finite collection of nilpotent elements. We prove that all Lie derived powers of $A$ are finitely generated Lie algebras.