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SUMMARY:Paolo Saracco (Université Libre de Bruxelles)
DTSTART:20230522T140000Z
DTEND:20230522T150000Z
DTSTAMP:20260423T035739Z
UID:EQuAL/12
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/EQuAL/12/">C
 losed categories\, modules and (one-sided) Hopf algebras</a>\nby Paolo Sar
 acco (Université Libre de Bruxelles) as part of European Quantum Algebra 
 Lectures (EQuAL)\n\n\nAbstract\nA well-known characterization of Hopf alge
 bras\, that I always found fascinating and elegant\, states that an algebr
 a A over a field k is a Hopf algebra if and only if its category of module
 s is a closed monoidal category in such a way that the forgetful functor t
 o vector spaces preserves the closed monoidal structure. We usually split 
 this result into two steps: the lifting of the monoidal structure correspo
 nds to the bialgebra structure\, and then the further lifting of the close
 d structure as adjoint to the monoidal one corresponds to the existence of
  an antipode. However\, closed structures can be defined independently of 
 monoidal ones and have their own dignity and importance. Which new structu
 re on our algebra A would correspond to lifting the closed structure of ve
 ctor spaces alone? How would this relate with the familiar bialgebra and H
 opf algebra structures coming from lifting the monoidal and closed monoida
 l ones? It turns out that lifting the closed structure corresponds to the 
 existence of algebra maps 𝛿 : A -> A⊗A^op and ε : A -> k satisfying 
 appropriate conditions. Moreover\, a quite unexpected source of examples i
 s provided by certain one-sided Hopf algebras\, i.e. bialgebras with a mor
 phism which is just a one-sided convolution inverse of the identity. In th
 is seminar\, based on an ongoing collaboration with Johannes Berger and Jo
 ost Vercruysse which is continuing discussions with Gabriella Böhm\, I wi
 ll present our progresses in the study of these new algebraic structures.\
 n
LOCATION:https://researchseminars.org/talk/EQuAL/12/
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