In this paper some results on the structure of finite-dimensional Lie algebras are obtained by means of the concept of maximal abelian dimension. More concretely, a sufficient condition is given for the solvability in finite-dimensional Lie algebras by using maximal abelian dimensions. Besides, a necessary condition for the nilpotency is also stated for such Lie algebras. Finally, the maximal a...

A cohomology theory of weighted Rota-Baxter $3$-Lie algebras is introduced. Formal deformations, abelian extensions, skeletal 2-algebras and crossed modules 3-Lie are interpreted by using lower degree groups.

A Lie version of Turaev’s G-Frobenius algebras from 2-dimensional homotopy quantum field theory is proposed. The foundation for this Lie version is a structure we call a g-quasi-Frobenius Lie algebra for g a finite dimensional Lie algebra. The latter consists of a quasi-Frobenius Lie algebra (q, β) together with a left g-module structure which acts on q via derivations and for which β is g-inva...

In these notes, we give a brief overview of the (finite dimensional) representation theory of finite dimensional semisimple Lie algebras. We first study the example of sl2(C) and then provide the general (additive) theory, along with an analysis of the representations of sl3(C). In the last section, we have a look at the multiplicative structure of the representation ring, discussing examples f...

Introduction This purpose of this paper is to summarize research done in the theory of Lie algebras, with attention directed towards contractions of Lie algebras. It is written in the style of an introduction to these topics for undergraduates familiar with (though not necessarily proficient in) linear and abstract algebra. It is separated into three sections: section (1) describes the basic de...

Results on characterization of manifolds in terms of certain Lie algebras growing on them, especially Lie algebras of differential operators, are reviewed and extended. In particular, we prove that a smooth (realanalytic, Stein) manifold is characterized by the corresponding Lie algebra of linear differential operators, i.e. isomorphisms of such Lie algebras are induced by the appropriate class...

A study is made of real Lie algebras admitting a hypersymplectic structure, and we provide a method to construct such hypersymplectic Lie algebras. We use this method in order to obtain the classification of all hypersymplectic structures on fourdimensional Lie algebras, and we describe the associated metrics on the corresponding Lie groups. MSC. 17B60, 53C15, 53C30, 53C50.

Let g be a complex reductive Lie algebra. We investigate a formalism for producing a natural model for representations of sl(2) × g which decompose in such a way so as to produce interesting correspondences between representations of sl(2) and of g. Our formalism generalizes the standard dual pair SL(2)× O(2n). We use it to obtain a formally unitary representation of sl(2,R)×u(3) that essential...