The projective indecomposable modules for the restricted Zassenhaus algebras in characteristic 2

It is shown that for the restricted Zassenhaus algebra $\mathfrak{W}=\mathfrak{W}(1,n)$, $n>1$, defined over an algebraically closed field $\mathbb{F}$ of characteristic 2 any projective indecomposable restricted $\mathfrak{W}$-module has maximal possible dimension $2^{2^n-1}$, and thus is isomorphic to some induced module $\mathrm{ind}^{\mathfrak{W}}_{\mathfrak{t}}(\mathbb{F}(\mu))$ for some torus of maximal dimension $\mathfrak{t}$. This phenomenon is in contrast to the behavior of finite-dimensional simple restricted Lie algebras in characteristic $p>3$

A group theoretical version of Hilbert's theorem 90

It is shown that for a normal subgroup $N$ of a group $G$, $G/N$ cyclic, the kernel of the map $N^{\mathrm{ab}}\to G^{\mathrm{ab}}$ satisfies the classical Hilbert 90 property (cf. Thm. A). As a consequence, if $G$ is finitely generated, $|G:N|<\infty$, and all abelian groups $H^{\mathrm{ab}}$, $N\subseteq H\subseteq G$, are torsion free, then $N^{\mathrm{ab}}$ must be a pseudo permutation module for $G/N$ (cf. Thm. B). From Theorem A one also deduces a non-trivial relation between the order of the transfer kernel and co-kernel which determines the Hilbert-Suzuki multiplier (cf. Thm. C). Translated into a number theoretic context one obtains a strong form of Hilbert's theorem 94. In case that $G$ is finitely generated and $N$ has prime index $p$ in $G$ there holds a "generalized Schreier formula" involving the torsion free ranks of $G$ and $N$ and the ratio of the order of the transfer kernel and co-kernel (cf. Thm. D)

Split strongly abelian p-chief factors and first degree restricted cohomology

In this paper we investigate the relation between the multiplicities of split strongly abelian p-chief factors of finite-dimensional restricted Lie algebras and first degree restricted cohomology. As an application we obtain a characterization of solvable restricted Lie algebras in terms of the multiplicities of split strongly abelian p-chief factors. Moreover, we derive some results in the representation theory of restricted Lie algebras related to the principal block and the projective cover of the trivial irreducible module of a finite-dimensional restricted Lie algebra. In particular, we obtain a characterization of finite-dimensional solvable restricted Lie algebras in terms of the second Loewy layer of the projective cover of the trivial irreducible module. The analogues of these results are well known in the modular representation theory of finite groups.Comment: 10 pages. arXiv admin note: substantial text overlap with arXiv:1206.366

Outer restricted derivations of nilpotent restricted Lie algebras

In this paper we prove that every finite-dimensional nilpotent restricted Lie algebra over a field of prime characteristic has an outer restricted derivation whose square is zero unless the restricted Lie algebra is a torus or it is one-dimensional or it is isomorphic to the three-dimensional Heisenberg algebra in characteristic two as an ordinary Lie algebra. This result is the restricted analogue of a result of T\^og\^o on the existence of nilpotent outer derivations of ordinary nilpotent Lie algebras in arbitrary characteristic and the Lie-theoretic analogue of a classical group-theoretic result of Gasch\"utz on the existence of $p$-power automorphisms of $p$-groups. As a consequence we obtain that every finite-dimensional non-toral nilpotent restricted Lie algebra has an outer restricted derivation.Comment: 9 pages, minor revisions, to appear in Proc. Amer. Math. So

Split abelian chief factors and first degree cohomology for Lie algebras

In this paper we investigate the relation between the multiplicities of split abelian chief factors of finite-dimensional Lie algebras and first degree cohomology. In particular, we obtain a characterization of modular solvable Lie algebras in terms of the vanishing of first degree cohomology or in terms of the multiplicities of split abelian chief factors. The analogues of these results are well known in the modular representation theory of finite groups. An important tool in the proof of these results is a refinement of a non-vanishing theorem of Seligman for the first degree cohomology of non-solvable finite-dimensional Lie algebras in prime characteristic. As applications we derive several results in the representation theory of restricted Lie algebras related to the principal block and the projective cover of the trivial irreducible module of a finite-dimensional restricted Lie algebra. In particular, we obtain a characterization of solvable restricted Lie algebras in terms of the second Loewy layer of the projective cover of the trivial irreducible module.Comment: 12 pages; minor revision