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SUMMARY:Hopf formulas for cocommutative Hopf algebras
DTSTART:20260319T100000
DTEND:20260319T110000
DTSTAMP:20260427T154925Z
UID:6abdd2629b808d5a0e4847fa106bc4bc5e2939dfda7889a7d7b0d05c
CATEGORIES:Conferences - Seminars
DESCRIPTION:de Marino Gran\, UCLouvain\nIn recent years\, numerous new app
 lications of categorical Galois theory have emerged in various interesting
  non-abelian algebraic contexts. In particular\, within semi-abelian categ
 ories\, this approach has led to some new calculations of higher fundament
 al groups in terms of generalized commutators in categories such as that o
 f compact groups\, crossed modules\, and skew braces. These categories sha
 re some structural properties with the categories of groups and of Lie alg
 ebras\, and also with the category of cocommutative Hopf algebras over a f
 ield\, which is also semi-abelian.\nThis raises the natural question of wh
 ether similar homological methods can be applied to study cocommutative Ho
 pf algebras as well.\n\nIn this talk\, after reviewing some fundamental pr
 operties of semi-abelian categories and some motivating examples\, I will 
 explain that the answer to the above question is affirmative. By using the
  exactness properties of cocommutative Hopf algebras and the free functor 
 universally associating a Hopf algebra with any coalgebra it is possible t
 o establish some new Hopf-type formulae for the homology of cocommutative 
 Hopf algebras. An important role is played by cleft extensions\, namely th
 ose surjective morphisms of Hopf algebras that are split as coalgebra morp
 hisms.\nWith any cleft extension\, one can associate a 5-term exact sequen
 ce in homology that can be seen as a Hopf-theoretic analogue of the classi
 cal Stallings-Stammbach exact sequence in group theory. This new approach 
 can also be applied to investigate the homology of cocommutative Hopf brac
 es\, which are interesting structures that naturally occur in the study of
  solutions to the so-called quantum Yang-Baxter equation. The category of 
 cocommutative Hopf braces turns out to be both semi-abelian and monadic on
  the category of coalgebras\, so that it is possible to investigate it fro
 m the perspective of non-abelian homological algebra.\n\nThis talk is base
 d on a joint work with Andrea Sciandra.\n 
LOCATION:CM 1 517 https://plan.epfl.ch/?room==CM%201%20517
STATUS:CONFIRMED
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