BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Paolo Saracco (University of Seville\, Spain) DTSTART:20250915T150000Z DTEND:20250915T160000Z DTSTAMP:20260423T035639Z UID:ENAAS/140 DESCRIPTION:Title: Integrals for bialgebras\nby Paolo Saracco (University of Seville\, Sp ain) as part of European Non-Associative Algebra Seminar\n\n\nAbstract\nAn extremely familiar notion in Hopf algebra theory is that of an \\emph{int egral}. If $G$ is a compact topological group\, then a Haar integral on $G $ is a linear functional $\\mathcal{C}(G) \\to \\mathbb{R}$ which is trans lation invariant.\n%\nA well-known result by Larson and Sweedler shows tha t integrals on a Hopf algebra can be obtained by applying the Structure Th eorem of Hopf modules to the rational part of its linear dual\, i.e.\, the space of integrals comes from a right adjoint functor from a category of modules to the category of vector spaces. \n%\nIn this seminar we will dis cuss how this construction can be carried out even in the absence of an an tipode\, offering a novel perspective on integrals which differs profoundl y from their classical description as ``colinear forms''. Our approach lea ds to a new notion of integrals for bialgebras which does not require\, an d does not imply\, the existence of an antipode.\n\nTime permitting\, we w ill seize this opportunity to say a few words about Hopf envelopes of bial gebras\, since in many cases of interest integrals for a bialgebra are in bijection with integrals for its Hopf envelope.\n%\nThe Hopf envelope of a bialgebra B is a certain universal Hopf algebra that we can associate wit h B and that plays for it the same role that the universal enveloping grou p plays for a monoid. \n%In categorical terms\, the Hopf envelope is the l eft adjoint to the forgetful functor from Hopf algebras to bialgebras\, he nce it may be legitimately called the free Hopf algebra generated by B. \n Its existence is a well-known fact in Hopf algebra theory\, but its constr uction is very technical. Nevertheless\, there are a number of cases where we can realize the Hopf envelope of a bialgebra B as a suitable quotient of B itself and we can take advantage of it to study integrals for the cor responding bialgebra.\n\n\\medskip\n\n\\centering \nThis talk is based on an ongoing project with A.\\ Ardizzoni and C.\\ Menini.\n LOCATION:https://researchseminars.org/talk/ENAAS/140/ END:VEVENT END:VCALENDAR