On the derived algebra of a centraliser

Let g be a classical Lie algebra, e∈g a nilpotent element and ge⊂g the centraliser of e. We prove that ge=[ge,ge] if and only if e is rigid. It is also shown that if e∈[ge,ge], then the nilpotent radical of ge coincides with [ge(1),ge], where ge(1)⊂ge is an eigenspace of a characteristic of e corresponding to the eigenvalue 1.     

Lie derived length and involutions in group algebras

Let $G$ be a group such that the set of $p$-elements of $G$ forms a finite nonabelian subgroup, where $p$ is an odd prime, and let $F$ be a field of characteristic $p$. In this paper we prove that the lower bound of the Lie derived length of the group algebra $FG$ given by Shalev in [11] is also a lower bound for the Lie derived length of the set of symmetric elements of $FG$ for every involution which is linear extension of an involutive anti-automorphism of $G$. Furthermore, we provide counterexamples to the interesting cases which are not covered by the main theorem.     

On fully invariant ideals of the free group algebra

Let be the group algebra of a free noncyclic group over an integral domain . It is proved that if is not a field, then there exists a fully invariant ideal of such is torsion-free but not projective as an -module. In other words, there exists a pure nonprojective variety of group representations over .

     

On the first Hochschild cohomology group of an algebra

Let A≡KΔ /I be a factor of a path algebra. We develop a strategy to compute dim H ¹(A), the dimension of the first Hochschild cohomology group of A, using combinatorial data from (Δ,I). That allows us to connect dim H ¹(A) with the rank and p-rank of the fundamental group π₁(Δ,I) of (Δ,I). We get explicit formulae for dim H ¹(A), when every path in Δ parallel to an arrow belongs to I or when I is homogeneous. Received: 12 April 1999 / Revised version: 9 October 2000     

On the group of modular units and the ideal class group

J.-M. Kim, S. Bae and I.-S. Lee showed that there exists an isomorphism between the p-primary part of the ideal class group and p-primary part of the unit group modulo cyclotomic unit group in Q⁺(ζpn) for all sufficiently large n under some conditions. In the present paper, we shall give an analogue of their result for modular units.     

On the derived length of solvable Lie algebras

Supported by NSF grant DMS-8501514     

On the length of the spectral sequence of a Lie algebra extension

The length of the spectral sequence of a Lie algebra extension is at most the dimension of the quotient algebra. We show that this bound can be attained for arbitrarily large quotient algebras even when the algebra is nilpotent and the extension splits.