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Resume Nous utilisons des resultats classiques sur les zeros des fonctions L de Dirichlet pour prouver la non-nullite de deux determinants analogues au determinant de Maillet. Nous en deduisons un theoreme d'extension pour les isometries de Lee et euclidienne des codes lineaires sur un corps premier.

Let G be a connected reductive algebraic group and let G ′ = [ G , G ] be its derived subgroup. Let ( G , V ) be a multiplicity free representation with a one-dimensional quotient (see definition below). We prove that the algebra D ( V ) G ′ of G ′ -invariant differential operators with polynomial coefficients on V , is a quotient of a so-called Smith algebra over its center. Over C this class of algebras was introduced by S.P. Smith (1990) as a class of algebras similar to U ( sl 2 ) . Our result generalizes the case of the Weil representation, where the associative algebra generated by Q ( x ) and Q ( ∂ ) ( Q being a non-degenerate quadratic form on V ) is a quotient of U ( sl 2 ) . Other structure results are obtained when ( G , V ) is a regular prehomogeneous vector space of commutative parabolic type. To cite this article: H. Rubenthaler, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

The object of this Note is to define the generalized flag geometry of a graded Lie algebra which corresponds to the generalized projective geometry in the case of 3-gradings. Then we construct a structure of manifold on this generalized flag geometry. This result generalizes a result known for 3-graded Lie algebras to the more general case of ( 2 k + 1 ) -graded Lie algebras. To cite this article: J. Chenal, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

The aim of this paper is to present remarkable classes of Lie-admissible algebras containing in particular the associative algebras, the Vinberg algebras and pre-Lie algebras. We determine the associated quadratic operads and their dual operads., Comment: 5 pages

We study the Lie algebra structure of the first Hochschild cohomology group of a finite dimensional monomial algebra A, in terms of the combinatorics of its quiver, in any characteristic. This allows us also to examine the identity component of the algebraic group of outer automorphisms of A in characteristic zero. Criteria for the (semi-)simplicity, the solvability, the reductivity, the commutativity and the nilpotency are given., 22 pages Correction of typing errors