• Citer

Split extensions, Actions and Crossed Modules in the categories of Hopf algebras

par Sterck, Florence
Président du jury Willem, Michel
Promoteur Vercruysse, Joost ;Gran, Marino
Publication Non publié, 2021-10-14

Thèse de doctorat

• Accès en ligne
• Détails
• Contenu
• Statistiques

Résumé :

This thesis develops and strengthens the interactions between two active research fields of mathematics: the theory of semi-abelian categories and the theory of Hopf algebras.On the one hand, we study the category of cocommutative Hopf algebras over any field and prove that this category is semi-abelian. Thanks to this result, we obtain a description of internal categorical objects in the Hopf algebras world, for example the Huq commutator, the internal crossed modules, and the internal crossed squares. We also obtain new results as the Zassenhaus Lemma for cocommutative Hopf algebras over any field. On the second hand, we focus on the category of general Hopf algebras in any symmetric monoidal category. We develop an adequate notion of split extensions of Hopf algebras (and bialgebras). This notion is adequate for several reasons. In particular, the Split Short Five Lemma holds when we restrict it to these extensions. Moreover, the split extensions of Hopf algebras (and bialgebras) are equivalent to the actions of Hopf algebras (and bialgebras). This equivalence allows us to obtain an equivalence between precrossed modules and S-reflexive graphs for general Hopf algebras. This thesis increases the links between Hopf algebras and category theory, providing a thorough study of the categorical notions of actions, precrossed modules, and crossed modules of Hopf algebras in various contexts.