Lie Properties of Restricted Enveloping Algebras

Let L be a restricted Lie algebra over a field of positive characteristic. We survey the known results about the Lie structure of the restricted enveloping algebra u(L) of L. Related results about the structure of the group of units and the symmetric and skew-symmetric elements of u(L) are also discussed. Moreover, a new theorem about an upper bound for the Lie nilpotency class of u(L) is proved

Enveloping algebras that are principal ideal rings

Let $L$ be a restricted Lie algebra over a field of positive characteristic. We prove that the restricted enveloping algebra of $L$ is a principal ideal ring if and only if $L$ is an extension of a finite-dimensional torus by a cyclic restricted Lie algebra

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

Solvability of Poisson algebras

Let $P$ be a Poisson algebra with a Lie bracket $\{, \}$ over a field \F of characteristic $p\geq 0$. In this paper, the Lie structure of $P$ is investigated. In particular, if $P$ is solvable with respect to its Lie bracket, then we prove that the Poisson ideal $\mathcal{J}$ of $P$ generated by all elements $\{\{\{x_1, x_2\}, \{x_3, x_4\}\}, x_5\}$ with $x_1,\ldots ,x_5 \in P$ is associative nilpotent of index bounded by a function of the derived length of $P$. We use this result to further prove that if $P$ is solvable and $p\neq 2$, then the Poisson ideal $\{P,P\}P$ is nil

Identities and derived lengths of finitary incidence algebras and their group of units

Let $FI(X,K)$ be the finitary incidence algebra of a poset $X$ over a field $K$. In this short note we establish when $FI(X,K)$ satisfies a polynomial identity and when its group of units $\mathcal{U}(FI(X,K))$ satisfies a group identity. The Lie derived length of $FI(X,K)$ and the derived length of $\mathcal{U}(FI(X,K))$ are also determined.Comment: Revised according to referee's comment

Getting to know you: Identification of pygmy killer whales (Feresa attenuata) and melon-headed whales (Peponocephala electra) under challenging conditions

Melon-headed whale (Peponocephala electra) and Pygmy killer whale (Feresa attenuata) are very poorly known species and are often confused with each other. We examined in detail Figure 3 in MARIGO and GIFFONI (2010) who reported that two melon-headed whales were taken in a surface driftnet about 90 nm off Santos, Brazil. We concluded they were in fact pygmy killer whales and explain our reasoning. To aid in future identifications, we illustrate and describe some of the main differences between these two species of small cetaceans. The incident reported by MARIGO and GIFFONI (2010) might represent the 'tip of the iceberg' regarding the incidental catches of cetaceans by pelagic drift nets off Brazil. Offshore driftnetting operating along the south-southeastern coast of Brazil may threaten pygmy killer whales.A orca-pigmeia (Feresa attenuata) era conhecida por poucos registros há não mais que 60 anos atrás, mas, apesar do número de registros ter crescido recentemente em todos os oceanos tropicais, F. attenuata é ainda considerada uma espécie pouco estudada. No Brasil, mesmo em base a um pequeno número de registros, presume-se sua distribuição como pelágica. Neste trabalho discutimos o registro de captura acidental de duas 'blackfish' (F. attenuata e Peponocephala electra) na costa norte de São Paulo, publicado na Figura 3 em MARIGO and GIFFONI (2010) e propomos a correção da identificação desses espécimes. A correta identificação dos três exemplares como orca-pigmeia coloca uma intrigante questão sobre a conservação dessa espécie no Atlântico Sul tropical. As operações de pesca com redes de deriva ao longo da costa sul-sudeste do Brasil podem ameaçar F. attenuata, espécie naturalmente rara. É recomendado o efetivo monitoramento da frota pesqueira, tendo em vista a necessidade de se avaliar a magnitude dessas capturas

MOSE: Salvare Venezia

Since 1900, the water levels in the Venetian Lagoon have risen almost 16 centimeters. In the same timeframe the city of Venice has gone from averaging 7 floods a year to over 100 in 2004 alone. It’s clear that in order for Venice to survive this problem a solution must be found. Fortunately, in 2003 MOSE (Modulo Sperimentale Elettromeccanico), a network of floodgates constructed at each entrance to the lagoon that prevents flooding during high tides, finally began its long awaited construction, that today is in its final stages. When looking at MOSE from an engineering perspective, the mechanics behind the system are actually quite simple. However, considering the magnitude of the project, they are incredibly important to understand thoroughly and to analyze correctly. This project seeks to understand the mechanical system that is currently in place, analyzing it in such a way in order to explain the strength of the floodgates