Generators of simple Lie algebras in arbitrary characteristics

In this paper we study the minimal number of generators for simple Lie algebras in characteristic 0 or p > 3. We show that any such algebra can be generated by 2 elements. We also examine the 'one and a half generation' property, i.e. when every non-zero element can be completed to a generating pair. We show that classical simple algebras have this property, and that the only simple Cartan type algebras of type W which have this property are the Zassenhaus algebras.Comment: 26 pages, final version, to appear in Math. Z. Main improvements and corrections in Section 4.

Slavery and the Revival of Anti-slavery Activism

This chapter sets out the volumes critical approach to the dominant discourse on modern slavery and its impulse to question the assumptions and the politics behind that discourse. It explores the limits of the modern slavery rhetoric for understanding the complicated logics of agency, freedom and belonging, and of past, present and future, for those who are constituted as slaves. Document type: Part of book or chapter of boo

A História da Alimentação: balizas historiográficas

Os M. pretenderam traçar um quadro da História da Alimentação, não como um novo ramo epistemológico da disciplina, mas como um campo em desenvolvimento de práticas e atividades especializadas, incluindo pesquisa, formação, publicações, associações, encontros acadêmicos, etc. Um breve relato das condições em que tal campo se assentou faz-se preceder de um panorama dos estudos de alimentação e temas correia tos, em geral, segundo cinco abardagens Ia biológica, a econômica, a social, a cultural e a filosófica!, assim como da identificação das contribuições mais relevantes da Antropologia, Arqueologia, Sociologia e Geografia. A fim de comentar a multiforme e volumosa bibliografia histórica, foi ela organizada segundo critérios morfológicos. A seguir, alguns tópicos importantes mereceram tratamento à parte: a fome, o alimento e o domínio religioso, as descobertas européias e a difusão mundial de alimentos, gosto e gastronomia. O artigo se encerra com um rápido balanço crítico da historiografia brasileira sobre o tema

Corps enveloppants des algèbres de Lie en dimension infinie et en caractéristique positive

Let g be a Lie algebra over a field k, U(g) its enveloping algebra, K(g) the ring of fractions of U(g). The aim of this thesis is to study algebraic properties of the division ring K(g) in the following two situations: on the one hand when k is of characteristic 0 and g is infinite-dimensional; on the other hand when k is of positive characteristic and g is finite-dimensional.Assume k is of characteristic 0. We first define the notion of 'transcendance degree of level q' for Poisson algebras. This notion is introduced from V. Petrogradsky's dimensions of level q for associative or Lie algebras. We prove, under mild assumptions on g, that the transcendance degree of level q+1 of K(g) is equal to the dimension of level q of g.We then proceed to the study of the family of Witt type Lie algebras defined by R. Yu. We can thus construct infinite families of pairwise non-isomorphic division rings with a given transcendance degree of level 3. We also study the question of centralisers in the enveloping skewfield of positive parts of Witt type algebras. In particular, we prove the following result: there exist infinite dimensional non-commutative Lie algebras g such that the first Weyl skewfield does not embed in K(g).Now assume that k is of characteristic p>0. We study the following specific Lie algebras: the matrix algebras gl(n); the algebras sl(n) provided p does not divide n; the modular Witt algebra W(1) and a subalgebra P of the Witt algebra W(2) (isomorphic to a tensor product of the Lie algebra W(1) with an associative algebra of truncated polynomials). In each case we prove that the enveloping skewfield is isomorphic to a Weyl skewfield. For the algebras W(1) and P, we also prove that the centre of the enveloping algebra is a unique factorisation domain, in agreement with a recent conjecture by A. Braun and C. Hajarnavis.Soient g une k-algèbre de Lie, U(g) son algèbre enveloppante, K(g) le corps des fractions de U(g). L'objet de cette thèse est d'étudier des propriétés algébriques du corps gauche K(g) dans les deux cas suivants : d'une part si k est de caractéristique 0 et g est de dimension infinie ; d'autre part si k est de caractéristique positive et g est de dimension finie.On suppose k de caractéristique nulle. On définit d'abord la notion de "degré de transcendance de niveau q" pour les algèbres de Poisson. Cette notion est introduite à partir de la notion de dimension de niveau q définie par V. Pétrogradsky pour les algèbres associatives et les algèbres de Lie. On démontre, sous des hypothèses peu restrictives sur g, que le degré de transcendance de niveau q+1 de K(g) est égal à la dimension de niveau q de g.On s'attache ensuite à l'étude de la famille des algèbres de type Witt définies par R. Yu. On construit ainsi des familles infinies de corps gauches deux à deux non isomorphes mais de même degré de transcendance de niveau 3 donné. On étudie aussi la question des centralisateurs dans les corps enveloppants des parties positives des algèbres de type Witt. On établit en particulier le résultat suivant : il existe des algèbres de Lie non commutatives de dimension infinie g telles que le premier corps de Weyl ne se plonge pas dans K(g).Supposons maintenant k de caractéristique p>0. On étudie le cas particuliers des algèbres de Lie suivantes : les algèbres gl(n) ; les algèbres sl(n) lorsque p ne divise pas n ; l'algèbre de Witt modulaire W(1) et une sous-algèbre P de l'algèbre de Witt W(2) (s'identifiant à un produit tensoriel de l'algèbre de Lie W(1) avec une algèbre associative de polynômes tronqués). Dans tous les cas, on démontre que le corps enveloppant est isomorphe à un corps de Weyl. Pour les algèbres W(1) et P, on démontre en outre que le centre de l'algèbre enveloppante est un anneau factoriel, en accord avec une conjecture récente de A. Braun et C. Hajarnavis