Weakly associative and symmetric Leibniz algebras

Abstract

We study a special class of weakly associative algebras: the symmetric Leibniz algebras. We describe the structure of the commutative and skew symmetric algebras associated with the polarization-depolarization principle. We also give a structure theorem for the symmetric Leibniz algebras and we study formal deformations in the context of deformation quantization.

Deformation quantization of nonassociative algebras

“…In fact the class of weakly associative algebras (associated to v = Id + c − τ 12 ) is the biggest class containing the Lie algebras and the associative algebras such that the v-deformation of a commutative associative algebra gives a Poisson algebra so quantizations of a Poisson algebra. In [13], we show also that the symmetric Leibniz algebras are also weakly associative.…”

“…Remark : Symmetric Leibniz algebras and weakly associative algebras. In [13] we have proved that symmetric Leibniz algebras and weakly associative algebras. If (A, µ 0 ) is a symmetric Leibniz algebra and if If we denote by A µ 0 the associator of the multiplication µ 0 , the first identity corresponds to…”