1,022 research outputs found

### 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)

### Rational discrete cohomology for totally disconnected locally compact groups

Rational discrete cohomology and homology for a totally disconnected locally
compact group $G$ is introduced and studied. The $\mathrm{Hom}$-$\otimes$
identities associated to the rational discrete bimodule $\mathrm{Bi}(G)$ allow
to introduce the notion of rational duality groups in analogy to the discrete
case. It is shown that semi-simple groups defined over a non-discrete,
non-archimedean local field are rational t.d.l.c. duality groups, and the same
is true for certain topological Kac-Moody groups. However, Y. Neretin's group
of spheromorphisms of a locally finite regular tree is not even of finite
rational discrete cohomological dimension. For a unimodular t.d.l.c. group $G$
of type $\mathrm{FP}$ it is possible to define an Euler-Poincar\'e
characteristic $\chi(G)$ which is a rational multiple of a Haar measure. This
value is calculated explicitly for Chevalley groups defined over a
non-discrete, non-archimedean local field $K$ and some other examples

### 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

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