Motivation. The theory of affine (Kac-Moody) Lie algebras has been a tremendous success story. Not only has one been able to generalize essentially all of the well-developed theory of finite-dimensional simple Lie algebras and their associated groups to the setting of affine Lie algebras, but these algebras have found many striking applications in other parts of mathematics. It is natural to as...

A Hom-algebra structure is a multiplication on a vector space where the structure is twisted by a homomorphism. The structure of Hom-Lie algebra was introduced by Hartwig, Larsson and Silvestrov in [4] and extended by Larsson and Silvestrov to quasi-hom Lie and quasi-Lie algebras in [5, 6]. In this paper we introduce and study Hom-associative, Hom-Leibniz, and Hom-Lie admissible algebraic struc...

The notion of pre-Lie algebra can be seen as a weakened form of associative algebra, still giving a Lie algebra by anti-symmetrization. It has been introduced by Gerstenhaber in his work on deformations of algebras [Ger64]. More recently, it has been studied from the point of view of operad theory and seen to be related to rooted trees [CL01]. The motivating problem for this paper is the classi...

Since 1870s, scientists have been taking deep insight into Lie groups and Lie algebras. With the development of Lie theory, Lie groups have got profound significance in many branches of mathematics and physics. In Lie theory, exponential mapping between Lie groups and Lie algebras plays a crucial role. Exponential mapping is the mechanism for passing information from Lie algebras to Lie groups....

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...

Every finite-dimensional Lie algebra is a semi-direct product of a solvable Lie algebra and a semisimple Lie algebra. Classifying the solvable Lie algebras is difficult, but the semisimple Lie algebras have a relatively easy classification. We discuss in some detail how the representation theory of the particular Lie algebra sl2 tightly controls the structure of general semisimple Lie algebras,...

We consider the 4D Martínez Alonso–Shabat equation E uty=uzuxy−uyuxz (also referred to as universal hierarchy equation) and using its known Lax pair construct two infinite-dimensional differential coverings over E. In these coverings, we give a complete description of Lie algebras nonlocal symmetries. particular, our results generalize ones obtained in Morozov Sergyeyev (2014) contain construct...

We initiate the study of ξ-groups and Hu-Liu Leibniz algebras, claim that almost all simple Leibniz algebras and simple Hu-Liu Leibniz algebras are linear, and establish two passages. One is the passage from a special Z2-graded associative algebra to a Hu-Liu Leibniz algebra. The other one is the passage from a linear ξ-group to its tangent space which is a Hu-Liu Leibniz algebra. The Lie corre...