ar X iv : m at h / 06 06 10 5 v 3 [ m at h . R A ] 2 5 Ja n 20 07 On the algebras obtained by tensor product
نویسنده
چکیده
where μA (resp. μB) is the multiplication of A (resp. B). We deduce that the ”natural” tensor product μA⊗B = μA ⊗ μB provides A⊗ B with a Lie-admissible algebra structure. In [2] we have defined special classes of Lie-admissible algebras with relations of definition determined by an action of the subgroupsGi of the 3-degree symmetric group Σ3. We obtain quadratic operads, denoted by Gi−Ass and in this family we find operads of Lie-admissible, associative, Vinberg and pre-Lie algebras. For these operads we have proved that for every P-algebra A and P -algebra B the tensor product A⊗B is a P-algebra. This is not true for general nonassociative algebras and, in this sense, the Gi-associative algebras are the most regular kind of nonassociative algebras. For example if P is the operad of Leibniz algebras or of the nonassociative algebras associated to Poisson algebras [?], then the tensor product of a P-algebra and P -algebra is not a P-algebra. So we introduce a quadratic operad, denoted by P̃ , such that the tensor product of a P-algebra with a P̃-algebra is a P-algebra. In case of P = Lie or Gi-Ass then P̃ = P ! this explain the above remarks.
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تاریخ انتشار 2006