On split extensions of bialgebras
(2018) Category Theory 2018 — Location: Ponta Delgada, Portugal (8.July.2018)
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- Abstract
- Inspired by the discovery in [3] that the category of cocommutative Hopf algebras over an algebraically closed field is semi-abelian, in the article [2] we prove a universal characterisation of Hopf algebras amongst cocommutative bialgebras. It is of the style of the characterisations of groups amongst monoids obtained in [4, 1]: a cocommutative bialgebra is a Hopf algebra precisely when every split extension over it admits a join decomposition. The aim of this talk is to give an account of some obstructions that occur when extending these results to bialgebras which are no longer cocommutative. References [1] X. García-Martínez, A new characterisation of groups amongst monoids, Appl. Categ. Structures 25 (2017), no. 4, 659–661. [2] X. García-Martínez and T. Van der Linden, A note on split extensions of bialgebras, Forum Math. (2018), in press. [3] M. Gran, G. Kadjo, and J. Vercruysse, A torsion theory in the category of cocommutative Hopf algebras, Appl. Categ. Structures 24 (2016), no. 3, 269–282. [4] A. Montoli, D. Rodelo, and T. Van der Linden, Two characterisations of groups amongst monoids, J. Pure Appl. Algebra 222 (2018), 747–777.
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Van der Linden, T. (2018). On split extensions of bialgebras. Category Theory 2018, Ponta Delgada, Portugal. https://hdl.handle.net/2078.5/61193