In this article, using the definitions of central series and nilpotency in the Lie algebras, we give some results similar to the works of Hulse and Lennox in 1976 and Hekster in 1986. Finally we will prove that every non trivial ideal of a nilpotent Lie algebra nontrivially intersects with the centre of Lie algebra, which is similar to Philip Hall's result in the group theory.

A C∞-Hopf algebra is a C∞-algebra which is also a convenient Hopf algebra with respect to the structure induced by the evaluations of smooth functions. We characterize those C∞-Hopf algebras which are given by the algebra C∞(G) of smooth functions on some compact Lie group G, thus obtaining an anti-isomorphism of the category of compact Lie groups with a subcategory of convenient Hopf algebras.

Let H be the quaternion algebra. Let g be a complex Lie algebra and let U(g) be the enveloping algebra of g. The quaternification g = (H ⊗ U(g), [ , ]gH ) of g is defined by the bracket [ z ⊗ X , w ⊗ Y ] gH = (z · w) ⊗ (XY ) − (w · z) ⊗ (Y X) , for z, w ∈ H and the basis vectors X and Y of U(g). Let SH be the ( non-commutative) algebra of H-valued smooth mappings over S and let Sg = SH ⊗ U(g). ...

One of the algebraic structures that has emerged recently in the study of the operator product expansions of chiral fields in conformal field theory is that of a Lie conformal algebra [K]. A Lie pseudoalgebra is a generalization of the notion of a Lie conformal algebra for which C[∂] is replaced by the universal enveloping algebra H of a finite-dimensional Lie algebra [BDK]. The finite (i.e., f...

Let $L$ be a finite-dimensional Lie algebra. We say a subalgebra $H$ of $L$ is permutably complemented in $L$ if there is a subalgebra $K$ of $L$ such that $L=H+K$ and $Hcap K=0$. Also, if every subalgebra of $L$ is permutably complemented in $L$, then $L$ is called completely factorisable. In this article, we consider the influence of these concepts on the structure of a Lie algebra, in partic...

In this work we consider 2-step nilradicals of parabolic subalgebras the simple Lie algebra An and describe a new family faithful nil-representations na,c, a,c ? N. We obtain sharp upper bound for minimal dimension ?(na,c) several pairs (a,c) ?(na,c).

There are also four appendices. Let K be a field of characteristic 0, and let C be a commutative K-algebra. which makes C into a Lie algebra, and is a biderivation (i.e. a derivation in each argument). The pair C, {−, −} is called a Poisson algebra. Poisson brackets arise in several ways. Example 1.1. Classical Hamiltonian mechanics. Here K = R, X is an even dimensional differentiable manifold ...

We study exceptional torsion in the integral cohomology of a family of p-groups associated to p-adic Lie algebras. A spectral sequence E r [g] is defined for any Lie algebra g which models the Bockstein spectral sequence of the corresponding group in characteristic p. This spectral sequence is then studied for complex semisimple Lie algebras like sln(C), and the results there are transferred to...

A reductive Lie algebra g is one that can be written C(g) ⊕ [g,g], where C(g) denotes the center of g. Equivalently, for any ideal a, there is another ideal b such that g = a⊕ b. A Cartan subalgebra of g is a subalgebra h that is maximal with respect to being abelian and having ad X being semisimple for all X ∈ h. For a reductive group, h = C(g) ⊕ h′, where h′ is a Cartan subalgebra of the semi...