On Hopf 2-algebras

Yaël Frégier , Friedrich Wagemann (1)

1 LMJL - Laboratoire de Mathématiques Jean Leray

Our main goal in this paper is to translate the diagram relating groups, Lie algebras and Hopf algebras to the corresponding 2-objects, i.e. to categorify it. This is done interpreting 2-objects as crossed modules and showing the compatibility of the standard functors linking groups, Lie algebras and Hopf algebras with the concept of a crossed module. One outcome is the construction of an enveloping algebra of the string Lie algebra of Baez-Crans, another is the clarification of the passage from crossed modules of Hopf algebras to Hopf 2-algebras.

Mots clés: crossed modules of groups; Lie algebras; Hopf algebras; Hopf 2-algebras; Lie 2-algebras; 2-groups; string group

Domaines: Théorie des groupes [math.GR]; Catégories et ensembles [math.CT]; Algèbres quantiques [math.QA]; Topologie algébrique [math.AT]

Yaël Frégier, Friedrich Wagemann. On Hopf 2-algebras. International Mathematics Research Notices, 2011, 2011, pp.3471-3501. ⟨10.1093/imrn/rnq224⟩. ⟨hal-00410695⟩