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                <authority type="supervisor">Rosane Ushirobira</authority>
                <authority type="supervisor">Georges Pinczon</authority>
                <authority type="jury">Didier Arnal [Président]</authority>
                <authority type="jury">Saïd Benayadi [Rapporteur]</authority>
                <authority type="jury">Rupert Yu [Rapporteur]</authority>
                <authority type="jury">Lucy Moser-Jauslin</authority>
                <authority type="jury">Andrea Solotar</authority>
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                <term xml:lang="en">Double extensions</term>
                <term xml:lang="en">Jordan-admissible</term>
                <term xml:lang="en">Generalized double extension</term>
                <term xml:lang="en">T*-extension</term>
                <term xml:lang="en">2-step nilpotent</term>
                <term xml:lang="en">Solvable Lie algebras</term>
                <term xml:lang="en">Lie algebra sp(2n)</term>
                <term xml:lang="en">Lie algebra o(m)</term>
                <term xml:lang="en">Adjoint orbits</term>
                <term xml:lang="en">Invariant</term>
                <term xml:lang="en">Symmetric Novikov algebras</term>
                <term xml:lang="en">Pseudo-Eucliean Jordan algebras</term>
                <term xml:lang="en">Quadratic Lie superalgebras</term>
                <term xml:lang="en">Quadratic Lie algebras</term>
                <term xml:lang="fr">Pas de mot clé en français</term>
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              <p>In this thesis, we defind a new invariant of quadratic Lie algebras and quadratic Lie superalgebras and give a complete study and classification of singular quadratic Lie algebras and singular quadratic Lie superalgebras, i.e. those for which the invariant does not vanish. The classification is related to adjoint orbits of Lie algebras o(m) and sp(2n). Also, we give an isomorphic characterization of 2-step nilpotent quadratic Lie algebras and quasi-singular quadratic Lie superalgebras for the purpose of completeness. We study pseudo-Euclidean Jordan algebras obtained as double extensions of a quadratic vector space by a one-dimensional algebra and 2-step nilpotent pseudo-Euclidean Jordan algebras, in the same manner as it was done for singular quadratic Lie algebras and 2-step nilpotent quadratic Lie algebras. Finally, we focus on the case of a symmetric Novikov algebra and study it up to dimension 7.</p>
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              <p>Dans cette thèse, nous définissons un nouvel invariant des algèbres de Lie quadratiques et des superalgèbres de Lie quadratiques et donnons une étude et classification complète des algèbres de Lie quadratiques singulières et des superalgèbres de Lie quadratiques singulières, i.e. celles pour lesquelles l’invariant n’est pas nul. La classification est en relation avec les orbites adjointes des algèbres de Lie o(m) et sp(2n). Aussi, nous donnons une caractérisation isomorphe des algèbres de Lie quadratiques 2-nilpotentes et des superalgèbres de Lie quadratiques quasi-singulières pour le but d’exhaustivité. Nous étudions les algèbres de Jordan pseudoeuclidiennes qui sont obtenues des extensions doubles d’un espace vectoriel quadratique par une algèbre d’une dimension et les algèbres de Jordan pseudo-euclidienne 2-nilpotentes, de la même manière que cela a été fait pour les algèbres de Lie quadratiques singulières et des algèbres de Lie quadratiques 2-nilpotentes. Enfin, nous nous concentrons sur le cas d’une algèbre de Novikov symétrique et l’étudions à dimension 7.</p>
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