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Habilitation à diriger des recherches

Antiautomorphismes d'algèbres et objets reliés.

Abstract : This report concerns the study of algebra antiautomorphisms, the case of linear antiautomorphisms over central simple algebras is specially concerned. When the algebra is a matrix algebra, such an antiautomorphism is the adjonction for a bilinear form. Hence, the classification of linear antiautomorphisms is a generalization of the classification of bilinear forms (up to similarity). In a first part, the asymmetry of a sesquilinear form is defined, and we give a characterization of the elements in a matrix algebra which are asymmetries. The notion of product of sesquilinear forms gives rise to a Morita theory for algebras with antiautomorphism. We then define the orthogonal sum of such algebras which are Morita equivalent with asymmetry. In a second part, after some remarks about the power of the the asymmetry in the classification of bilinear forms over fields, we explain how the notion of asymmetry is generalized to linear antiautomorphismes over central simple algebras. We give a list of questions and comments about what we could hope using this asymmetry. The study of the Hasse principle for similarities of bilinear forms naturally leads to the computation of some Tate-Shafarevich groups of some normic algebraic torus. This allows us in a third part to produce some counter-examples to this hasse princile, and to give an interpretation of the obstruction to this principle in terms of class field theory. Some other computations of Tate-Shavarevich groups for algebraic tori prove that those tori are not stably rational. This result gives the simple algebraic groups which generic torus is rational, and restricts the cases where the rationality of the generic torus could give the rationality of the group. The fourth part is the definition and the study of some natural and functorial ionvariants, generalizing the known invariants for involutions (and quadratic forms) : discriminant, clifford algebra, trace form. We develop some classification results obtained in small dimension and explain the results we could hope in small cohomological dimension.
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Habilitation à diriger des recherches
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Submitted on : Monday, July 5, 2010 - 6:08:11 PM
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Anne Cortella. Antiautomorphismes d'algèbres et objets reliés.. Mathématiques [math]. Université de Franche-Comté, 2010. ⟨tel-00497746⟩

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