Ernesto Spinelli

Group algebras whose symmetric and skew elements are Lie solvable

Abstract Let FG be the group algebra of a group G without 2-elements over a field F of characteristic p ≠ 2 endowed with the canonical involution induced from the map g ↦ g –1, g ∈ G. Let (FG)– and (FG)+ be the sets of skew and symmetric elements of FG, respectively, and let P denote the set of p-elements of G (with P = 1 if p = 0). In the present paper we prove that if either P is finite or G is non-torsion and (FG)– or (FG)+ is Lie solvable, then FG is Lie solvable. The remaining cases are also settled upon small restrictions.

Lie nilpotency indices of restricted universal enveloping algebras

ABSTRACT For a restricted Lie algebra L over a field of characteristic p > 0 we study the Lie nilpotency index t L (u(L)) of its restricted universal enveloping algebra u(L). In particular, we determine an upper and a lower bound for t L (u(L)). Finally, under the assumption that L is p-nilpotent and finite-dimensional, we establish when the Lie nilpotency index of u(L) is maximal. Communicated by I. Shestakov.

Group Rings Whose Symmetric Units are Solvable

Let K be an infinite field of characteristic different from 2, and G a group. Under suitable restrictions upon G, we classify the groups such that the symmetric units of KG satisfy the solvability identity (x 1, x 2,…, x 2 n ) o = 1, for some n.

Lie Nilpotent Group Algebras and Upper Lie Codimension Subgroups

In this article we introduce the series of the upper Lie codimension subgroups of a group algebra KG of a group G over a field K. By means of this series we give a contribution to the conjecture cl L (KG) = cl L (KG) when G belongs to particular classes of finite p-groups.

On the derived length of the unit group of a group algebra

Abstract Let KG be a non-commutative group algebra of a torsion nilpotent group G over a field K of positive characteristic p whose unit group, 𝒰(KG), is solvable. In the present note we prove that dl(𝒰(KG)) ⩾ ⌈log2(p + 1)⌉ and characterize group algebras for which this lower bound is achieved.

Group Rings Whose Symmetric Units Generate an n-Engel Group

Let F be an infinite field of characteristic different from 2 and G a torsion group. Write 𝒰+(FG) for the set of units in the group ring FG that are symmetric with respect to the classical involution induced from the map g ↦ g −1, for all g ∈ G. We classify the groups such that ⟨𝒰+(FG)⟩ is n-Engel.

Lie dimension subgroups and central series related to group algebras

Let KG be the group algebra of a group G over a field K of positive characteristic p, and let 𝔇(n)(G) and 𝔇[n](G) denote the n-th upper Lie dimension subgroup and the n-th lower one, respectively. In [1] and [12], the equality 𝔇(n)(G) =𝔇[n](G) is verified when p ≥ 5. Motivated by [16, Problem 55], in the present paper we establish it for particular classes of groups when p ≤ 3. Finally, we introduce and study a new central series of G linked with the Lie nilpotency class of KG.