A note on split extensions of bialgebras
-
Xabier García-Martínez
and
Tim Van der Linden
Abstract
We prove a universal characterization of Hopf algebras among cocommutative bialgebras over an algebraically closed field: a cocommutative bialgebra is a Hopf algebra precisely when every split extension over it admits a join decomposition. We also explain why this result cannot be extended to a non-cocommutative setting.
Funding source: Xunta de Galicia
Award Identifier / Grant number: GRC2013-045
Funding source: Ministerio de Economía y Competitividad
Award Identifier / Grant number: MTM2016-79661-P
Funding statement: This work was partially supported by Ministerio de Economía y Competitividad (Spain), grant MTM2016-79661-P (AEI/FEDER, UE, support included). The first author was also supported by Xunta de Galicia, grant GRC2013-045 (European FEDER support included), by an FPU scholarship of the Ministerio de Educación, Cultura y Deporte (Spain) and by a Fundación Barrié scholarship. The second author is a Research Associate of the Fonds de la Recherche Scientifique–FNRS.
Acknowledgements
Thanks to Georgios Charalambous, José Manuel Fernández Vilaboa, Isar Goyvaerts, Marino Gran, James R. A. Gray, Gabriel Kadjo and Joost Vercruysse for fruitful discussions and useful comments. We would also like to thank the University of Cape Town and Stellenbosch University for their kind hospitality during our stay in South Africa.
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© 2018 Walter de Gruyter GmbH, Berlin/Boston
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- A note on split extensions of bialgebras
- Petersson norm of cusp forms associated to real quadratic fields
- A realization theorem for sets of lengths in numerical monoids
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- Heat kernel estimates for time fractional equations
- Extreme non-Arens regularity of the group algebra
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- Global integrability for solutions to some anisotropic problem with nonstandard growth
- Hardy operators on Musielak–Orlicz spaces
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- Path homology theory of multigraphs and quivers
Articles in the same Issue
- Frontmatter
- Upper bounds for geodesic periods over rank one locally symmetric spaces
- Additive bases with coefficients of newforms
- A note on split extensions of bialgebras
- Petersson norm of cusp forms associated to real quadratic fields
- A realization theorem for sets of lengths in numerical monoids
- On the non-existence of the Mackey topology for locally quasi-convex groups
- The Davies method revisited for heat kernel upper bounds of regular Dirichlet forms on metric measure spaces
- The cup product of Brooks quasimorphisms
- Heat kernel estimates for time fractional equations
- Extreme non-Arens regularity of the group algebra
- Rational homology and homotopy of high-dimensional string links
- Global integrability for solutions to some anisotropic problem with nonstandard growth
- Hardy operators on Musielak–Orlicz spaces
- Contractibility of the stability manifold for silting-discrete algebras
- Order of the canonical vector bundle over configuration spaces of spheres
- An endpoint version of uniform Sobolev inequalities
- Space-time L2 estimates, regularity and almost global existence for elastic waves
- k-spaces and duals of non-archimedean metrizable locally convex spaces
- Path homology theory of multigraphs and quivers
