arXiv subjects
Publications and source records attributed to Michel Goze.
In this work, we are interested in a non symmetric homogeneous space, namely $SO(2m)/Sp(m)$. We show that this space admits a structure of $Z_2^2$-symmetric space. We describe all the non degenerated metrics and classify the Riemannian and Lorentzian ones.
Let P be a quadratic operad. We determine an associated operad ~P such that for any P-algebra A and any ~P-algebra B then the tensor product $A \otimes B$ is a P-algebra.
In the first section we recall some basic notions on Lie algebras. In a second time we study the algebraic variety of complex $n$-dimensional Lie algebras. We present different notions of deformations : Gerstenhaber deformations, pertubations, valued deformations and we use these tools to study some properties of this variety. Finaly we introduce the concept of rigidity and we present some results on the class of rigid Lie algebras.
A current Lie algebra is contructed from a tensor product of a Lie algebra and a commutative associative algebra of dimension greater than 2. In this work we are interested in deformations of such algebras and in the problem of rigidity. In particular we prove that a current Lie algebra is rigid if it is isomorphic to a direct product gxg...xg where g is a rigid Lie algebra.
In this work we study riemannian metrics on flag manifolds adapted to the symmetries of these homogeneous nonsymmetric spaces. We first introduce the notion of riemannian $Γ$-symmetric space when $Γ$ is a general abelian finite group, the symmetric case corresponding to $Γ=\Z_2$. We describe and study all the riemannian metrics on $SO(2n+1)/SO(r_1)\times SO(r_2)\times SO(r_3)\times SO(2n+1-r_1-r_2-r_3)$ for which the symmetries are isometries. We consider also the lorentzian case and give an example of a lorentzian homogeneous space which is not a symmetric space.
We present coalgebras of some classes of nonassociative algebras whose associator satisfies invariance conditions given by the action of the 3-order symmetric group. Amongst these algebras we find the well-known Vinberg algebras, the Pre-Lie algebras, the Lie-admissible algebras and the 3-power associative algebras. We study properties between these coalgebras and algebras which generalize the associative case.
We present classes of nonassociative algebras whose associator satisfies invariance conditions given by the action of the 3 order symmetric group. Amongst these algebras we find the wellknown Pre Lie or Vinberg algebras and the Lie admissible algebras. We study the corresponding noncoassociative cogebras.
The complete affine structures on abelian Lie algebras in small dimensions are well known. In this paper we are interested by the non complete case. In particular we classify all these structures in dimensions 2 and 3.
The eThe existence of common zero of a family of polynomials has led to study of inertial forms whose homogeneous part of degree 0 constitutes the ideal resultant. The Koszul anc Cech cohomology groups play a fundamental role in this study. An analogueous of Hurwitz theorem is given and also we finds a MCoy theorem in a particular case of this study.
We develop the notion of deformations using a valuation ring as ring of coefficients. This permits to consider in particular the classical Gerstenhaber deformations of associative or Lie algebras as infinitesimal deformations and to solve the equation of deformations in a polynomial frame. We consider also the deformations of the enveloping algebra of a rigid Lie algebra and we define valued deformations for some classes of non associative algebras.
A Lie-admissible algebra gives by anticommutativity a Lie algebra. In this work we study remarkable classes of Lie-admissible algebras such as Vinberg, PreLie algebras. We compute the corresponding binary quadratic operads and study their Koszul duality. Considering Lie algebras as Lie-admissible algebras we can define for each Lie algebra a cohomology with values in a Lie-admissible module. This permits to study some deformations of Lie algebras in the category of Lie-admissible algebras. Lastly we study the tensor product between these operads and their dual operads. As application we construct new classes of Vinberg algebras.
We give some examples of non-complete invariant affine connections on nilpotent and filiform Lie groups. This permits to describe non-nilpotent faithful representations on the model of filiform n-dimensional Lie algebras and, in particular, on the 3 dimensional Heisenberg algebra.