Related papers: Representing Lie algebras using approximations wit…
We prove an analog of the Ado theorem - the existence of a finite-dimensional faithful representation - for a certain kind of finite-dimensional nilpotent Hom-Lie algebras.
We give a simple proof of the Birkhoff theorem about existence of a faithful representation for any finite-dimensional nilpotent Lie algebra of characteristic zero.
In this paper, we construct a faithful representation with the lowest dimension for every complex Lie algebra in dimension $\leq 4$. In particular, in our construction, in the case that the faithful representation has the same dimension of…
We give a natural proof of the Ado theorem.
A class of representations of a Lie superalgebra (over a commutative superring) in its symmetric algebra is studied. As an application we get a direct and natural proof of a strong form of the Poincare'-Birkhoff-Witt theorem, extending this…
We introduce the concept of a triangular representation of a Lie algebra, give a counterpart of Ado's theorem, and discuss $2$-irreducible triangular modules over a nonreductive Lie algebra.
Ado's Theorem had been extended to principal ideal domains independently by Churkin and Weigel. They demonstrated that if $R$ is a principal ideal domain of characteristic zero and $\mathfrak{L}$ is a Lie algebra over $R$ which is also a…
We describe a new method to determine faithful representations of small dimension for a finite dimensional nilpotent Lie algebra. We give various applications of this method. In particular we find a new upper bound on the minimal dimension…
We prove a finite torsion-free associative conformal algebra to have a finite faithful conformal representation. As a corollary, it is shown that one may join a conformal unit to such an algebra. Some examples are stated to demonstrate that…
In the article at hand, we sketch how, by utilizing nilpotency to its fullest extent (Engel, Super Engel) while using methods from the theory of universal enveloping algebras, a complete description of the indecomposable representations may…
Let L be a restricted Lie algebra over a field of positive characteristic. We survey the known results about the Lie structure of the restricted enveloping algebra u(L) of L. Related results about the structure of the group of units and the…
A Lie algebra $L$ is said to be of breadth $k$ if the maximal dimension of the images of left multiplication by elements of the algebra is $k$. In this paper we give characterization of finite dimensional nilpotent Lie algebras of breadth…
The paper is devoted to the study of pro-solvable Lie algebras whose maximal pro-nilpotent ideal is either $\mathfrak{m}_0$ or $\mathfrak{m}_2$. Namely, we describe such Lie algebras and establish their completeness. Triviality of the…
In this paper we present a lower bound for the minimal dimension $\mu(\mathfrak{n})$ of a faithful representation of a finite dimensional $p$-step nilpotent Lie algebra $\mathfrak{n}$ over a field of characteristic zero. Our bound is given…
In this paper we find minimal faithful representations of several classes filiform Lie algebras by means of strictly upper-triangular matrices. We investigate Leibniz algebras whose corresponding Lie algebras are filiform Lie algebras such…
In this note we establish several basic properties of nilpotent Sabinin algebras. Namely, we show that nilpotent Sabinin algebras (1) can be integrated to produce nilpotent loops, (2) satisfy an analogue of the Ado theorem, (3) have…
The main purpose of this paper is to study the finite-dimensional solvable Lie algebras described in its title, which we call {\em minimal non-${\mathcal N}$}. To facilitate this we investigate solvable Lie algebras of nilpotent length $k$,…
In this paper, we introduce near perfect ideals and upper bounded ideals, and study them as well as perfect ideals for finite dimensional Lie algebras. We show that the largest perfect ideal and the largest near perfect ideal of a finite…
The paper extends existing Lie algebra representation theory related to Lie algebra gradings. The notion of a representation compatible with a given grading is defined and applied to finite-dimensional representations of $sl(n,\mathbb{C})$…
For a representation of a Lie algebra, one can construct a diagram of the representation, i. e. a directed graph with edges labeled by matrix elements of the representation. This article explains how to use these diagrams to describe normal…