Let K be a field of positive characteristic p and KG the group algebra of a group G. It is known that, if KG is Lie nilpotent, then its upper (or lower) Lie nilpotency index is at most |G| + 1, where |G| is the order of the commutator subgroup. Previously we determined the groups G for which the upper/lower nilpotency index is maximal or the upper nilpotency index is ‘almost maximal’ (that is, ...

Let K be a field of positive characteristic p and KG the group algebra of a group G. It is known that, if KG is Lie nilpotent, then its upper (or lower) Lie nilpotency index is at most |G|+ 1, where |G| is the order of the commutator subgroup. The authors have previously determined the groups G for which this index is maximal and here they determine the G for which it is ‘almost maximal’, that ...

Given a space X let LX denote its rational homotopy Lie algebra π∗(ΩX) ⊗ Q. A cell attachment f : ∨iSi → X is said to be free if the Lie ideal in LX generated by f is a free Lie algebra. This condition is shown to be general in the following sense. Given a space X with rational cone length N , then X is rationally homotopy equivalent to a space constructed using at most N + 1 free cell attachme...

A possible notion of superspace associated to a given 3-dimensional Lie algebra g0 might be a Lie superalgebra g g0 ⊕ g1. If one is to introduce a minimal set of assumptions, it seems quite natural to consider those superalgebras for which g1 g0 and for which the g0 action on g1 is the adjoint representation. A complete classification of the real and complex 3, 3 -dimensional Lie superalgebras ...

The paper presents a subclass of the class of MD5-algebras and MD5-groups, i.e., five dimensional solvable Lie algebras and Lie groups such that their orbits in the co-adjoint representation (K-orbit) are orbit of zero or maximal dimension. The main results of the paper is the classification up to an isomorphism of all MD5-algebras G with the derived ideal G1 := [G,G] is a 3-dimensional commuta...

Let F be a free Lie algebra of finite rank over a field K. We prove that if an ideal [Formula: see text] of the algebra [Formula: see text] contains a primitive element [Formula: see text] then the element [Formula: see text] is primitive. We also show that, in the Lie algebra [Formula: see text] there exists an element [Formula: see text] such that the ideal [Formula: see text] contains a prim...

Let R be a 2-torsion free σ-prime ring. It is shown here that if U 6⊂ Z(R) is a σ-Lie ideal of R and a, b in R such that aUb = σ(a)Ub = 0, then either a = 0 or b = 0. This result is then applied to study the relationship between the structure of R and certain automorphisms on R. To end this paper, we describe additive maps d : R −→ R such that d(u) = 2ud(u) where u ∈ U, a nonzero σ-square close...

In this article we prove that for a basic classical Lie superalgebra the annihilator of a strongly typical Verma module is a centrally generated ideal. For a basic classical Lie superalgebra of type I we prove that the localization of the enveloping algebra by a certain central element is free over its centre.

‎let $l$ be a lie algebra‎, ‎$mathrm{der}(l)$ be the set of all derivations of $l$ and $mathrm{der}_c(l)$ denote the set of all derivations $alphainmathrm{der}(l)$ for which $alpha(x)in [x,l]:={[x,y]vert yin l}$ for all $xin l$‎. ‎we obtain an upper bound for dimension of $mathrm{der}_c(l)$ of the finite dimensional nilpotent lie algebra $l$ over algebraically closed fields‎. ‎also‎, ‎we classi...

The purpose of this paper is to describe a C program FPLSA for investigating finitely presented Lie algebras and superalgebras. The program takes as input data a finite set of generators and relations for these elements. The relations have the form of Lie polynomials with coefficients being either integers or polynomials over integers in a given finite set of parameters. The program is based on...