Related papers: Simple Lie Algebras having Extremal Elements

We show that a minimal ideal of a finite-dimensional Lie algebra is either simple or abelian.

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 line with known results on groups, we show the existence of the nilpotent radical in Lie rings with minimal condition on centralizers. We also prove a form of Engel's theorem if the characteristic is zero.

By a proper cover of a finite group G we mean an extension of a nontrivial finite group by G. Our purpose is to show that a proper cover of a finite simple group L of Lie type always contains an element whose order differs from the element…

We report on some computations with reachable elements in simple Lie algebras of exceptional type within the SLA package of GAP4. These computations confirm the classification of such elements by Elashvili and Grelaud. Secondly they answer…

We answer a conjecture of Bauer, Catanese and Grunewald showing that all finite simple groups other than the alternating group of degree 5 admit unmixed Beauville structures. We also consider an analog of the result for simple algebraic…

The class of minimal non-elementary Lie algebras over a field F are studied. These are classified when F is algebraically closed and of characteristic different from 2,3. The solvable algebras in this class are also characterised over any…

We study Lie algebra prederivations. A Lie algebra admitting a non-singular prederivation is nilpotent. We classify filiform Lie algebras admitting a non-singular prederivation but no non-singular derivation. We prove that any 4-step…

For sufficiently high dimensions, the naturally graded nonsplit nilpotent Lie algebras with linear characteristic sequence are classified.

In this note we prove that every non characteristically filiform Lie algebra is endowed with an affine structure.

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 this paper, we classify all capable nilpotent Lie algebras with the derived subalgebra of dimension 2 over an arbitrary field. Moreover, the explicit structure of such Lie algebras of class 3 is given.

The length of an element $z$ of a Lie algebra $L$ is defined as the smallest number $s$ needed to represent $z$ as a sum of $s$ brackets. The bracket width of $L$ is defined as supremum of the lengths of its elements. Given a…

Let $A$ be a split finite-dimensional associative unital algebra over a field. The first main result of this note shows that if the Ext-quiver of $A$ is a simple directed graph, then $HH^1(A)$ is a solvable Lie algebra. The second main…

We show that every unimodular Lie algebra, of dimension at most 4, equipped with an inner product, possesses an orthonormal basis comprised of geodesic elements. On the other hand, we give an example of a solvable unimodular Lie algebra of…

We prove that if a completely non-unitary contraction T in L(H) has a non-trivial algebraic element h, then T has a non-trivial invariant subspace.

Given two elements $A,B$ in a compact semisimple Lie algebra, we show that there is a regular element $X$ and elements $Y,Z$ with $A=[X,Y]$ and $B=[X,Z]$. In the course of the proof we show also that every linear subspace $V$ of codimension…

Let $K$ be an algebraically closed field of characteristic zero and $A$ an integral $K$-domain. The Lie algebra $Der_{K}(A)$ of all $K$-derivations of $A$ contains the set $LND(A)$ of all locally nilpotent derivations. The structure of…

We investigate the existence of left-invariant closed G$_2$-structures on seven-dimensional non-solvable Lie groups, providing the first examples of this type. When the Lie algebra has trivial Levi decomposition, we show that such a…

We show that semi-simple lie algebras can be characterized by their maximal nilpotent subalgebra, which is the same as the nilpotent radical of a Borel subalgebra.