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We establish some relations between quadratic symplectic Lie superalgebras and Manin superalgebras. Next, we introduce some concepts of double extensions of quadratic symplectic Lie superalgebras and of Manin superalgebras in order to give inductive descriptions of quadratic symplectic Lie superalgebras.

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We introduce the class of split Lie–Rinehart algebras as the natural extension of the one of split Lie algebras. We show that if [Formula: see text] is a tight split Lie–Rinehart algebra over an associative and commutative algebra [Formula: see text] then [Formula: see text] and [Formula: see text] decompose as the orthogonal direct sums [Formula: see text] and [Formula: see text], where any [Formula: see text] is a nonzero ideal of [Formula: see text], any [Formula: see text] is a nonzero ideal of [Formula: see text], and both decompositions satisfy that for any [Formula: see text], there exists a unique [Formula: see text] such that [Formula: see text]. Furthermore, any [Formula: see text] is a split Lie–Rinehart algebra over [Formula: see text]. Also, under mild conditions, it is shown that the above decompositions of [Formula: see text] and [Formula: see text] are by means of the family of their, respective, simple ideals.

International audienceA study of Leibniz bialgebras arising naturally through the double of Leibnizalgebras analogue to the classical Drinfeld’s double is presented. A key ingredient of ourwork is the fact that the underline vector space of a Leibniz algebra becomes a Lie algebraand also a commutative associative algebra, when provided with appropriate new products.A special class of them, the coboundary Leibniz bialgebras, gives us the natural frame-work for studying the Yang-Baxter equation (YBE) in our context, inspired in the classicalYang-Baxter equation as well as in the associative Yang-Baxter equation. Results of theexistence of coboundary Leibniz bialgebra on a symmetric Leibniz algebra under certainconditions are obtained. Some interesting examples of coboundary Leibniz bialgebras arealso included. The final part of the paper is dedicated to coboundary Leibniz bialgebrastructures on quadratic Leibniz algebras

Communicated by C. Kassel MSC: 17A70 17B05 17B20 a b s t r a c t The aim of this paper is to exhibit some non-trivial examples of quadratic Lie superalgebras such that the even part is a reductive Lie algebra and the action of the even part on the odd part is not completely reducible and to give an inductive description of this class of quadratic Lie superalgebras. The notion of the generalized double extension of quadratic Lie superalgebras proposed by I. Bajo, S. Benayadi and M. Bordemann [I. Bajo, S. Benayadi, M. Bordemann, Generalized double extension and descriptions of quadratic Lie superalgebras, 2007, arXiv:math-ph/0712.0228] is of crucial importance in this work.In particular, we will improve some results of [S. Benayadi, Quadratic Lie superalgebras with the completely reducible action of the even part on the odd part, J. Algebra 223 (2000) 344-366], in the sense that we will not demand that the action of the even part on the odd part is completely reducible, which naturally makes the proofs of our results more difficult.

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