Cat¹-Hopf Algebras and Crossed Modules

Abstract

We formulate the concept of crossed module of Hopf algebras in a symmetric monoidal category 𝒞. This notion generalizes the one given by Whitehead (1948) for crossed modules of groups. We characterize this new structure by introducing a definition of cat¹-Hopf algebra which is, in some sense, a dual notion of bigebroid given by Maltsiniotis (1992).

Key Words:

• Crossed module
• Crossed product
• Hopf algebra

2000 Mathematics Subject Classification:

• 18D10
• 16W30

ACKNOWLEDGMENTS

We are grateful to the referee for valuable comments and helpful suggestions that made the current version more readable. The authors were partially supported by Ministerio de Ciencia y Tecnología and by FEDER funds: Project BMF2003-07353-C02-01.

Notes

Communicated by R. Wisbauer.