On 3-Lie algebras with a derivation

2018

In this article, we introduce a new cohomology theory associated to a Lie 2-algebras. This cohomology theory is shown to extend the classical cohomology theory of Lie algebras; in particular, we show that the second cohomology group classifies an appropriate type of extensions.

Deformation Theory of Algebras and Structures and Applications, 1988

Mediterranean Journal of Mathematics, 2017

In this paper we investigate in details derivations on trivial extension algebras. We obtain generalizations of both known results on derivations on triangular matrix algebras and a known result on first cohomology group of trivial extension algebras. As a consequence we get the characterization of trivial extension algebras on which every derivation is inner. We show that, under some conditions, a trivial extension algebra on which every derivation is inner has necessarily a triangular matrix representation. The paper starts with detailed study (with examples) of the relation between the trivial extension algebras and the triangular matrix algebras.

2019

Let $ 0 \rightarrow A\rightarrow L {\rightarrow} B \rightarrow 0 $ be an abelian extension of Lie algebras. In this paper, we construct certain exact sequences which relate derivations with the Lie algebra cohomology group $ H^{2}(B,A) $, and apply them to study extending derivations of $ A $ and lifting derivations of $ B $ to certain derivations of $ L $.

Journal of Lie theory

Using the generalised notion of the Lie algebra of derivations, we introduce the second non-abelian cohomology of Lie algebras with coe-cients in crossed mod- ules and extend the seven-term exact non-abelian cohomology sequence of Guin to nine-term exact sequence. The second non-abelian cohomology of Lie algebras is described by Lie algebra extensions.

The author considers general questions of deformations of Lie algebras over a field of characteristic zero, and the related problems of computing cohomology with coefficients in adjoint representations. The construction of a versal family, and the construction of obstructions to the extension of deformations, are also considered. Bibliography: 13 titles.

BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS

Deformations of the three-dimensional Lie algebra sl(2) Deformation is one of key questions of the structural theory of algebras over a field. Especially, it plays a important role in the classification of such algebras. In odd characteristics of algebraically closed fields, local deformations of classical Lie algebras are completely described. Local deformations are also known for classical Lie algebras with a homogeneous root system over an algebraically closed field of characteristic 2, except for the three-dimensional Lie algebra sl(2). In the characteristic 2, deformations of Lie algebras with an non-homogeneous root system are calculated only for Lie algebras of small ranks. In this paper we investigate deformations of the three-dimensional classical Lie algebra sl(2) over an algebraically closed field k of characteristic p = 2. We also describe three-dimensional two-sided Alia algebras associated with Lie algebra sl(2) in the characteristics 2 and 3. It is proved that, in characteristic 2, the space of local deformations of the Lie algebra sl(2) is five-dimensional. The structural specialty of the second cohomology space of the adjoint representation of the Lie algebra sl(2) are analyzed. In particular, the subspace of cosets of restricted cocycles is described. It is proved that the subspace of classes of restricted cocycles is two-dimensional and the corresponding local deformations are restricted Lie algebras in the sense of Jacobson. It was found that a family of simple three-dimensional unrestricted Lie algebras correspond to unrestricted non-trivial cocycles. In characteristics 2 and 3, three-dimensional two-sided Alia algebras that are non-isomorphic to the Lie algebra sl(2) are constructed. In the process of the study, a complete description of the space of all derivations of the Lie algebra sl(2) is obtained.

2008

The existing constructions of derived Lie and sh-Lie brackets involve multilinear maps that are used to define higher order differential operators. In this paper, we prove the equivalence of three different definitions of higher order operators. We then introduce a unifying theme for building derived brackets and show that two prevalent derived Lie bracket constructions are equivalent. Two basic methods of constructing derived strict sh-Lie brackets are also shown to be essentially the same. So far, each of these derived brackets is defined on an abelian subalgebra of a Lie algebra. We describe, as an alternative, a cohomological construction of derived sh-Lie brackets. Namely, we prove that a differential algebra with a graded homotopy commutative and associative product and an odd, square-zero operator (that commutes with the differential) gives rise to an sh-Lie structure on the cohomology via derived brackets. The method is in particular applicable to differential vertex operato...

arXiv (Cornell University), 2019

In this paper, we first study derivations in non nilpotent Lie triple algebras. We determine the structure of derivation algebra according to whether the algebra admits an idempotent or a pseudo-idempotent. We study the multiplicative structure of non nilpotent dimensionally nilpotent Lie triple algebras. We show that when n = 2p + 1 the adapted basis coincides with the canonical basis of the gametic algebra G(2p + 2, 2) or this one obviously associated to a pseudo-idempotent and if n = 2p then the algebra is either one of the precedent case or a conservative Bernstein algebra.

1986

The author considers general questions of deformations of Lie algebras over a field of characteristic zero, and the related problems of computing cohomology with coefficients in adjoint representations. The construction of a versal family, and the construction of obstructions to the extension of deformations, are also considered. Bibliography: 13 titles. In this paper, we consider general questions on deformations of Lie algebras over a field of characteristic zero, and related problems of computing cohomology with coefficients in adjoint representations. We consider the construction of a versal family and the nature of obstructions to the extension of deformations. Our aim is to carry over general constructions of the modern theory of deformations and related properties of the cohomology of (local) commutative algebras to Lie algebras in parallel with the papers [3], [10], and [11]. 1. We shall require some information on the Harrison cohomology of commutative rings (see [12] and [...