A nonzero element x of a Lie algebra L over a field F is called extremal if [x, [x,L]] ⊆ Fx. Extremal elements are a well-studied class of elements in simple finite-dimensional Lie algebras of Chevalley type: they are the long root elements. In [CSUW01], Cohen, Steinbach, Ushirobira and Wales have studied Lie algebras generated by extremal elements, in particular those of Chevalley type. The au...

We introduce the notion of a Lie-like algebra⋄ (superalgebra⋄) for ⋄ ∈ { , , 3−rd }. By bundling a family of algebras, we introduce six new generalizations of Lie algebras which are called Lie-like algebras or Lie-like superalgebras in this paper. The examples of the new generalizations of Lie algebras come from an invariant algebra (or an invariant superalgebra), where the notion of an invaria...

In this paper, first we introduce the notion of a Reynolds operator on an n-Lie algebra and illustrate relationship between operators derivations algebra. We give cohomology theory study infinitesimal deformations using second group. Then NS-n-Lie algebras, which are generalizations both algebras n-pre-Lie algebras. show that gives rise to together with representation itself. Nijenhuis naturall...

An Einstein nilradical is a nilpotent Lie algebra, which can be the nilradical of a metric Einstein solvable Lie algebra. The classification of Riemannian Einstein solvmanifolds (possibly, of all noncompact homogeneous Einstein spaces) can be reduced to determining, which nilpotent Lie algebras are Einstein nilradicals and to finding, for every Einstein nilradical, its Einstein metric solvable ...

In the first two sections, we review the Block-Wilson-PremetStrade classification of restricted simple Lie algebras. In the third section, we compute their infinitesimal deformations. In the last section, we indicate some possible generalizations by formulating some open problems. 1. Restricted Lie algebras We fix a field F of characteristic p > 0 and we denote with Fp the prime field with p el...

Let K be a field of positive characteristic p and KG the group algebra of a group G. It is known that, if KG is Lie nilpotent, then its upper (or lower) Lie nilpotency index is at most |G| + 1, where |G| is the order of the commutator subgroup. Previously we determined the groups G for which the upper/lower nilpotency index is maximal or the upper nilpotency index is ‘almost maximal’ (that is, ...

Building upon the concepts of the matrix Lie group and the matrix Lie algebra, we explore the natural connections between the Lie groups and Lie algebras via the exponential map. We later introduce the matrix commutator as a Lie bracket operation to aid our investigation of Lie algebra representations, which we illustrate with the example of the adjoint representation on the special unitary group.

We define Lie groups and Lie algebras and show how invariant vector fields on a Lie group form a Lie algebra. We prove that this correspondence respects natural maps and discuss conditions under which it is a bijection. Finally, we introduce the exponential map and use it to write the Lie group operation as a function on its Lie algebra.

This paper begins by introducing the concept of a quasi-hom-Lie algebra, or simply, a qhl-algebra, which is a natural generalization of hom-Lie algebras introduced in a previous paper [14]. 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 nat...

We propose a notion of a quantum universal enveloping algebra for an arbitrary Lie algebra defined by generators and relations which is based on the quantum Lie operation concept. This enveloping algebra has a PBW basis that admits the Kashiwara crystalization. We describe all skew primitive elements of the quantum universal enveloping algebra for the classical nilpotent algebras of the infinit...