A Hom-Lie algebra is a triple (L, [−,−], α), where α is a linear self-map, in which the skew-symmetric bracket satisfies an α-twisted variant of the Jacobi identity, called the Hom-Jacobi identity. When α is the identity map, the Hom-Jacobi identity reduces to the usual Jacobi identity, and L is a Lie algebra. Hom-Lie algebras and related algebras were introduced in [1] to construct deformation...

We introduce a notion of Hopf-Lie-Rinehart algebra and show that the universal algebra of a Hopf-Lie-Rinehart algebra acquires an ordinary Hopf algebra structure. Subject classification: Primary 16W30, Secondary 16S32 17B35

The paper is devoted to an application of Lie group theory to differential equations. The basic infinitesimal method for calculating symmetry group is presented, and used to determine general symmetry group of some differential equations. We include a number of important applications including integration of ordinary differential equations and finding some solutions of partial differential equa...

Let g denote a Lie algebra, and let ĝ denote the tensor product of g with a ring of truncated polynomials. The Lie algebra ĝ is called a truncated current Lie algebra. The highest-weight theory of ĝ is investigated, and a reducibility criterion for the Verma modules is described. Let g be a Lie algebra over a field k of characteristic zero, and fix a positive integer N . The Lie algebra (1) ĝ =...

We describe the ‘Lie algebra of classical mechanics’, modelled on the Lie algebra generated by kinetic and potential energy of a simple mechanical system with respect to the canonical Poisson bracket. It is a polynomially graded Lie algebra, a class we introduce. We describe these Lie algebras, give an algorithm to calculate the dimensions cn of the homogeneous subspaces of the Lie algebra of c...

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...

We study Novikov algebras and Novikov structures on finite-dimensional Lie algebras. We show that a Lie algebra admitting a Novikov structure must be solvable. Conversely we present an example of a nilpotent 2-step solvable Lie algebra without any Novikov structure. We construct Novikov structures on certain Lie algebras via classical r-matrices and via extensions. In the latter case we lift No...

(identical representation). The even part G0 is characterized by the condition a0i = ai0 = 0 for i = 1, . . . , n. G0 is isomorphic to gl(n). We choose the Cartan subalgebra H as the set of diagonal matrices of G and G0 resp. Let eij be denote the (n+ 1)× (n+ 1)-matrix with 1 at the place (i, j) and 0 everywhere (i, j = 0, . . . , n) . Then we get by h1 = e00 + e11 , hi = ei−1,i−1 − eii (i = 2,...

A nonzero element x of a Lie algebra L over a field F is called extremal if [x, [x,L]] ⊆ Fx. Extremal elements are a well-studied class of elements in simple finite-dimensional Lie algebras of Chevalley type: they are the long root elements. In [CSUW01], Cohen, Steinbach, Ushirobira and Wales have studied Lie algebras generated by extremal elements, in particular those of Chevalley type. The au...

Let P be a Poisson homogeneous G-space. In [Dr2], Drinfeld shows that corresponding to each p ∈ P , there is a maximal isotropic Lie subalgebra lp of the Lie algebra d, the double Lie algebra of the tangent Lie bialgebra (g, g∗) of G. Moreover, for g ∈ G, the two Lie algebras lp and lgp are related by lgp = Adg lp via the Adjoint action of G on d. In particular, they are isomorphic as Lie algeb...