Deformation quantization of non associative algebras

We study deformation quantization of nonassociative algebras whose associator satisfies some symmetric relations. This study is expanded to a larger class of nonassociative algebras includind Leibniz algebras. We apply also to this class the rule of polarization-depolarization.Comment: 19 page

Associative and Lie deformations of Poisson algebras

Considering a Poisson algebra as a non associative algebra satisfying the Markl-Remm identity, we study deformations of Poisson algebras as deformations of this non associative algebra. This gives a natural interpretation of deformations which preserves the underlying associative structure and we study deformations which preserve the underlying Lie algebra.Comment: 20 page

3-dimensional algebras. Part 1. Skew-symmetric case

An algebra is called skew-symmetric if its multiplication operation is a skew-symmetric bilinear application. We determine all these algebras in dimension $3$ over a field of characteristic different from $2$. As an application, we determine the subvariety of $3$-dimensional Hom-Lie algebras. For this type of algebras, we study also the dimension $4$.Comment: 18 page

Breadth and characteristic sequence of nilpotent Lie algebras

The notion of breadth of a nilpotent Lie algebra was introduced by B. Khuhirun, K.C. Misra and E. Stitzinger and used to approach problems of classification up to isomorphism. In the present paper, we study this invariant in terms of characteristic sequence, another invariant introduced by M. Goze and J.M. Ancochea-Bermudez. This permits to complete the determination of Lie algebras of breadth 2 and to begin the work for Lie algebras with breadth greater than 2.Comment: 12 page

Affine structures on filiform Lie algebras

In this note we prove that every non characteristically filiform Lie algebra is endowed with an affine structure.Comment: 6 page

Affine structures on nilpotent contact Lie algebras

In this paper we study some affine structures on nilpotent Lie algebras endowed with a contact form. These affine structures are constructed from an affine structure on a symplectic Lie algebra by a central extension.Comment: 9 page

Non-Existence of Complex Structures on Filiform Lie Algebras

The aim of this work is to prove the nonexistence of complex structures over nilpotent Lie algebras of maximal class (also called filiform).Comment: 13 page

A class of nonassociative algebras

In this paper we consider all these nonassociative algebras defined by the action of invariant subspaces of the symmetric group $\Sigma_3$ on the associator of the considered laws.Comment: 24 page

Universal deformation formulas

We give a conceptual explanation of universal deformation formulas for unital associative algebras and prove some results on the structure of their moduli spaces. We then generalize universal deformation formulas to other types of algebras and their diagrams.Comment: 22 page

Lie-admissible algebras and operads

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.Comment: 20 page