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Hopf formulae for cocommutative Hopf algebras
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DOI：10.1007/s10231-026-01665-5.png)
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Abstract
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The adjunction between coalgebras and Hopf algebras, first described by Takeuchi, allows one to prove that the semi-abelian category of cocommutative Hopf algebras has enough E-projective objects, where E is the class of cleft extensions. One then proves that, for any cocommutative Hopf algebra, there exists a weak E-universal normal (=central) extension. This fact allows one to apply the methods of categorical Galois theory to classify normal E-extensions and to provide an explicit description of the fundamental group of a cocommutative Hopf algebra in terms of a generalized Hopf formula. Moreover, with any cleft extension, we associate a 5-term exact homology sequence that can be seen as a Hopf-theoretic analogue of the classical Stallings-Stammbach exact sequence in group theory.
Keywords:
Hopf algebras
Semi-abelian categories
Categorical Galois theory
Hopf formulae
Baer invariants
Cleft extensions
Journal
A
IF:
0.9
Papers:
75
Citations:
0
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Cited Papers