Sur les bialgèbres faibles et les algèbres de Hopf faibles.

Abstract

In this thesis, we study the structure of weak bialgebras and weak Hopf algebras. These structures appeared in Physics, in particular in renormalization in quantum fields theory and q-deformations of oscillator algebras. They are generalization of bialgebras and Hopf algebras obtained by relaxing the condition that the comultiplication and the counit are algebra maps with respect to the unit element. In this work, we provide a complete study of weak bialgebra and weak Hopf algebras structures. We recall definitions and properties. Moreover a classification in dimension 2 and 3 are given. The main results deal with Kaplansky-type constructions. Indeed we show various constructions providing weak bialgebras and weak Hopf algebras, starting by an associative algebra. A second part of this thesis is dedicated to twisted structures, which are Hom-type algebras. We introduce weak hom-bialgebras and weak Hom-Hopf algebras for which we obtain similar results. In the last part of the thesis, we explore Ore extensions and deformations of weak bialgebras and weak Hopf algebras.