Extensions of Lie algebras (original) (raw)

D G ] 2 6 Fe b 20 04 EXTENSIONS OF LIE ALGEBRAS

2008

We review (non-abelian) extensions of a given Lie algebra, identify a 3-dimensional cohomological obstruction to the existence of extensions. A striking analogy to the setting of covariant exterior derivatives, curvature, and the Bianchi identity in differential geometry is spelled out.

Non-abelian cohomology and extensions of Lie algebras

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.

Extensions of super Lie algebras

2001

We study (non-abelian) extensions of a super Lie algebra and identify a cohomological obstruction to the existence, parallel to the known one for Lie algebras. An analogy to the setting of covariant exterior derivatives, curvature, and the Bianchi identity in differential geometry is shown.

A cohomology theory for Lie 2-algebras

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.

On 3-Lie algebras with a derivation

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.

Derivations of abelian Lie algebra extensions

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) ,andapplythemtostudyextendingderivationsof, and apply them to study extending derivations of ,andapplythemtostudyextendingderivationsof A $ and lifting derivations of $ B $ to certain derivations of $ L $.

Lie algebra extensions and abelian monopoles

Physics Letters B, 1987

By considering an appropriate exact sequence of Lie algebras and using properties of Lie algebra extensions, we give an abstract algebraic description of the abelian monopole. We show how a sequence of Lie algebras carries a notion of a covariant derivative. Finally, using a metric on the Lie algebras we discuss the abstract algebraic version of electromagnetism.

A cohomology theory for Lie 2-algebras and Lie 2-groups

arXiv: Differential Geometry, 2018

In this thesis, we introduce a new cohomology theory associated to a Lie 2-algebras and a new cohomology theory associated to a Lie 2-group. These cohomology theories are shown to extend the classical cohomology theories of Lie algebras and Lie groups in that their second groups classify extensions. We use this fact together with an adapted van Est map to prove the integrability of Lie 2-algebras anew.

Isoclinic extensions of Lie algebras

Turkish Journal of Mathematics

In this article we introduce the notion of the equivalence relation, isoclinism, on the central extensions of Lie algebras, and determine all central extensions occurring in an isoclinism class of a given central extension. We also show that under some conditions, the concepts of isoclinism and isomorphism between the central extensions of finite dimensional Lie algebras are identical. Finally, the connection between isoclinic extensions and the Schur multiplier of Lie algebras are discussed.