BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Kenichi Shimizu DTSTART:20210707T080000Z DTEND:20210707T090000Z DTSTAMP:20260415T175218Z UID:AdvancesHopfAlgebroids/1 DESCRIPTION:Title: FRT type construction of Hopf algebroids\nby Kenichi Sh imizu as part of Advances in Hopf Algebroids\n\n\nAbstract\nThe Faddeev-Re shetikhin-Takhtajan (FRT) construction associates a coquasitriangular bi- or Hopf algebra to a solution of the Yang-Baxter equation. There are vario us generalizations of this construction. What is important in this talk am ong them is Hayashi's generalization. He established a way of constructing a weak Hopf algebra from a Yang-Baxter face model\, which is\, mathematic ally\, a braided object in the category of bimodules over the finite produ ct of the base field. Being motivated by his work\, and also hoping applic ations to dynamical quantum groups\, I investigated category-theoretic asp ects of the FRT construction. In this talk\, I will introduce a way to con struct a bialgebroid over an algebra A from a strong monoidal functor from a small monoidal category\, D\, to the category of rigid A-bimodules. The resulting bialgebroid\, B\, may be called the FRT bialgebroid over A sinc e the category of B-comodules is lax braided if D is. Furthermore\, it tur ns out that B is a Hopf algebroid if the monoidal category D is rigid. The construction and the proof of these results are based on the fact that a bialgebroid over A is identified with a left adjoint bimonad on the catego ry of A-bimodules. The FRT bialgebroid is\, in fact\, obtained as a specia l case of a Tannaka theoretic construction of coend bimonads.\n LOCATION:https://researchseminars.org/talk/AdvancesHopfAlgebroids/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Zoran Škoda DTSTART:20210707T091500Z DTEND:20210707T101500Z DTSTAMP:20260415T175218Z UID:AdvancesHopfAlgebroids/2 DESCRIPTION:Title: Twisting Hopf algebroids and beyond\nby Zoran Škoda as part of Advances in Hopf Algebroids\n\n\nAbstract\nXu has introduced Drin feld twists for associative bialgebroids.\n\nGeneral proof that appropriat e formulas twist the antipode as well has been missing for 20 years.\n\nI shall sketch my result from 2019 which resolves the question if the antipo de is invertible. Then I will discuss twisting examples for Hopf algebroid s of differential operators and a relation of these examples to multiplica tive unitaries.\n\nProper treatment requires extensions of the formalism t o internal Hopf algebroids in monoidal categories with completed tensor pr oducts (after works of Böhm and Stojić). Further examples suggest going beyond Hopf algebroids to a nonassociative version or other examples towar d nonaffine quantum groupoids\, which are both a work in progress.\n LOCATION:https://researchseminars.org/talk/AdvancesHopfAlgebroids/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Daniel Schäppi DTSTART:20210707T120000Z DTEND:20210707T130000Z DTSTAMP:20260415T175218Z UID:AdvancesHopfAlgebroids/3 DESCRIPTION:Title: From homology theories to Hopf algebroids - and back!\n by Daniel Schäppi as part of Advances in Hopf Algebroids\n\n\nAbstract\nA n important source of examples of Hopf algebroids is a construction due to Adams which starts with a suitable homology theory and constructs a Hopf algebroid\, together with a lift of the homology theory to the category of comodules of that Hopf algebroid.\n\nThis construction only works if the homology theory satisfies a certain flatness condition. Pstragowski has fo und a way to describe the category of comodules as a category of additive sheaves on a site\, and this construction works in much greater generality . Moreover\, if one can show by abstract means that the resulting category of sheaves is equivalent to the category of comodules of some Hopf algebr oid\, then one obtains a new homology theory which does satisfy the flatne ss condition.\n\nThis motivates the following recognition question: when i s a category equivalent to the category of comodules of a Hopf algebroid? I will explain some general recognition results and how they can be applie d to the category constructed by Pstragowski.\n LOCATION:https://researchseminars.org/talk/AdvancesHopfAlgebroids/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Niels Kowalzig DTSTART:20210707T131500Z DTEND:20210707T141500Z DTSTAMP:20260415T175218Z UID:AdvancesHopfAlgebroids/4 DESCRIPTION:Title: A noncommutative calculus on the cyclic dual of Ext\nby Niels Kowalzig as part of Advances in Hopf Algebroids\n\n\nAbstract\nWe s how that if the cochain complex computing Ext groups in the category of mo dules over a Hopf algebroid admits a cocyclic structure\, then the noncomm utative Cartan calculus structure on Tor over Ext dualises in a cyclic sen se to a calculus on Coext over Cotor. More precisely\, the cyclic duals of the chain resp. cochain spaces computing the two classical derived functo rs lead to complexes that compute the more exotic ones\, giving a cyclic o pposite module over an operad with multiplication that induce operations s uch as a Lie derivative\, a cap product (or contraction)\, and a (cyclic) differential\, along with higher homotopy operators defining a noncommutat ive Cartan calculus up to homotopy. In particular\, this allows to recover the classical Cartan calculus from differential geometry or the Chevalley -Eilenberg calculus for Lie(-Rinehart) algebras without any finiteness con ditions or the use of topological tensor products.\n LOCATION:https://researchseminars.org/talk/AdvancesHopfAlgebroids/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Xiao Han DTSTART:20210707T143000Z DTEND:20210707T153000Z DTSTAMP:20260415T175218Z UID:AdvancesHopfAlgebroids/5 DESCRIPTION:Title: On the gauge groups and bialgebroids\nby Xiao Han as pa rt of Advances in Hopf Algebroids\n\n\nAbstract\nFor starters\, we will re call the fundamental concept of a Hopf-Galois extension\, and instantiate it through quantum principal SU(2)-bundles with noncommutative seven-spher es as total spaces and noncommutative four-spheres as base spaces\, togeth er with monopole bundle over the quantum sphere.\n\nThen we will recall th e construction of the Ehresmann-Schauenburg bialgebroid of a Hopf-Galois e xtension\, which is a noncommutative analogue of the Ehresmann groupoid of a classical principal bundle. Next\, we will show that\, the gauge group of any Hopf Galois extension is isomorphic to the group of bisections of i ts Ehresmann-Schauenburg bialgebroid. Then\, we will prove that the group of bisections and the group of automorphisms of the bialgebroid form a cro ssed module. In particular\, we will consider and Galois objects (non-triv ial noncommutative principal bundles over a point)\, whose base-space suba lgebra is the ground field and the corresponding Ehresmann-Schauenburg bia lgebroid becomes a Hopf algebra. Examples will include monopole bundle and Galois objects of Taft algebras.\n LOCATION:https://researchseminars.org/talk/AdvancesHopfAlgebroids/5/ END:VEVENT END:VCALENDAR