Algebra of derivations of Lie algebras

2019

Let be a Lie algebra and be two ideals of . If denotes the set of all derivations of whose images are in and send to zero, then we give necessary and sufficient conditions under which is equal to some special subalgebras of the derivation algebra of . We also consider finite dimensional Lie algebra for which the center of the set of inner derivations, , is equal to the set of central derivations of , , and give a characterisation of such Lie algebras.

Vietnam journal of mathematics, 2023

Nilpotent Lie algebras with maximal subalgebras are considered. A bound on the dimension of these algebras as a function of their coclass is obtained. These algebras are then classified up to coclass two. The result is field dependent and several fields are considered.

Journal of Algebra, 2000

some applications of the main results to the study of functions f ∈ Hom L L such that f • µ or µ • f ∧ I L defines a Lie multiplication.

2021

In this paper, we study 3-Lie algebras with derivations. We call the pair consisting of a 3-Lie algebra and a distinguished derivation by the 3-LieDer pair. We define a cohomology theory for 3-LieDer pair with coefficients in a representation. We study central extensions of a 3-LieDer pair and show that central extensions are classified by the second cohomology of the 3-LieDer pair with coefficients in the trivial representation. We generalize Gerstenhaber’s formal deformation theory to 3-LieDer pairs in which we deform both the 3-Lie bracket and the distinguished derivation.

Journal of Algebra and Its Applications

This paper is devoted to the nilpotent finite-dimensional evolution algebras E with dim [Formula: see text] = dim E − 1. We describe the Lie algebra of derivations of these algebras. Moreover, in terms of these Lie algebras, we fully construct nilpotent evolution algebra with maximal index of nilpotency. Furthermore, this result allowed us fully characterize all local and 2-local derivations of the considered evolution algebras. Besides, all automorphisms and local automorphisms of these algebras are found.

Linear Algebra and its Applications, 2016

We prove that every local derivation on a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic zero is a derivation. We also give examples of finite-dimensional nilpotent Lie algebras L with dim L ≥ 3 which admit local derivations which are not derivations.

Linear Algebra and its Applications, 2015

We prove that every 2-local derivation on a finite-dimensional semi-simple Lie algebra L over an algebraically closed field of characteristic zero is a derivation. We also show that a finite-dimensional nilpotent Lie algebra L with dim L ≥ 2 admits a 2-local derivation which is not a derivation.

Proceedings of the American Mathematical Society, 2012

In this paper we prove that every finite-dimensional nilpotent restricted Lie algebra over a field of prime characteristic has an outer restricted derivation whose square is zero unless the restricted Lie algebra is a torus or it is one-dimensional or it is isomorphic to the three-dimensional Heisenberg algebra in characteristic two as an ordinary Lie algebra. This result is the restricted analogue of a result of Tôgô on the existence of nilpotent outer derivations of ordinary nilpotent Lie algebras in arbitrary characteristic and the Lie-theoretic analogue of a classical group-theoretic result of Gaschütz on the existence of p-power automorphisms of p-groups. As a consequence we obtain that every finite-dimensional non-toral nilpotent restricted Lie algebra has an outer restricted derivation.

2017

An (associative) algebra is a vector space over a field equipped with an associative, bilinear multiplication. By use of a new bilinear operation, any associative algebra morphs into a nonassociative abstract Lie algebra, where the new product in terms of the given associative product, is the commutator. The crux of this paper is to investigate the commutator as it pertains to the general linear group and its subalgebras. This forces us to examine properties of ring theory under the lens of linear algebra, as we determine subalgebras, ideals, and solvability as decomposed into an extension of abelian ideals, and nilpotency, as decomposed into the lower central series and eventual zero subspace. The map sending the Lie algebra L to a derivation of L is called the adjoint representation, where a given Lie algebra is nilpotent if and only if the adjoint is nilpotent. Our goal is to prove Engel’s Theorem, which states that if all elements of L are ad-nilpotent, then L is nilpotent.