352 research outputs found

### 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 hÃ¡ nÃ£o mais que 60 anos atrÃ¡s, mas, apesar do nÃºmero de registros ter crescido recentemente em todos os oceanos tropicais, F. attenuata Ã© ainda considerada uma espÃ©cie pouco estudada. No Brasil, mesmo em base a um pequeno nÃºmero de registros, presume-se sua distribuiÃ§Ã£o como pelÃ¡gica. Neste trabalho discutimos o registro de captura acidental de duas 'blackfish' (F. attenuata e Peponocephala electra) na costa norte de SÃ£o Paulo, publicado na Figura 3 em MARIGO and GIFFONI (2010) e propomos a correÃ§Ã£o da identificaÃ§Ã£o desses espÃ©cimes. A correta identificaÃ§Ã£o dos trÃªs exemplares como orca-pigmeia coloca uma intrigante questÃ£o sobre a conservaÃ§Ã£o dessa espÃ©cie no AtlÃ¢ntico Sul tropical. As operaÃ§Ãµes de pesca com redes de deriva ao longo da costa sul-sudeste do Brasil podem ameaÃ§ar F. attenuata, espÃ©cie naturalmente rara. Ã‰ 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. Itâ€™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

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