[Dottorcomp] Insalate di Matematica - E. Masut 17/05/2023

Mer 17 Maggio 2023 10:00:00 CEST

Dear all,

We remind you the upcoming talk of "Insalate di Matematica".
Here the details:

*Speaker:* *Elisabetta Masut* (Università degli Studi di Padova)

*Date and time:* *17th of May 2023, 4:15 pm (CET)*

*Information to attend*
*In presence:* The seminar will take place in room 3014, at the building
U5-Ratio, Università degli Studi di Milano Bicocca.
*Online**:* The seminar will be streamed via Webex platform at the
following link:
https://unimib.webex.com/unimib/j.php?MTID=ma9fac62549ae03a1196907bfff0a7057
 (Password: Insalate (46725283 from phones))

*Title**:* "Non-existence of integral Hopf orders for twists of
several simple groups of Lie type."

*Abstract*:
In 1975 Kaplansky listed 10 conjectures on Hopf algebras, which have been
the focus of a great deal of research. Some of these conjectures are still
unanswered. In particular, we will be interested on the sixth one, that is
a generalization of Frobenius theorem in the framework of representation
theory for Hopf algebras.
It states that, given a complex finite-dimensional semisimple Hopf
algebra [image:
H], the dimension of every irreducible representation of [image: H] divides
the dimension of [image: H].
Larson proved a weaker version of it: if the complex finite-dimensional
semisimple Hopf algebra admits a Hopf order over a number ring, then
Kaplansky' sixth conjecture is satisfied.
A natural question now arises: does every complex
semisimple finite-dimensional Hopf algebra which satisfies Kaplansky'
sixth conjecture admit a Hopf order over a number ring?
The answer is negative.
In this talk, after a brief excursus on Hopf algebras, we present families
of Hopf algebras, which satisfy Kamplansky' sixth conjecture, but they do
not admit a Hopf order over a number ring. These Hopf algebras are
constructed as Drinfel'd twist of the group algebras [image: KG], with [image:
K] a number field. We will prove that for every finite simple group [image:
G], there is always a deformation, such that the twisted Hopf algebra does
not admit a Hopf order over a number ring. Moreover, we will show that for
two families of groups, this non-existence result holds for any twist.
This talk is based on a joint work with Giovanna Carnovale and Juan Cuadra.

*Keywords: *Hopf algebras, Hopf orders, Drinfel'd twist.

*** We inform you that the talk will be recorded and uploaded on our
website. If you join the talk after the starting time, we kindly ask you to
ensure that your microphone and webcam are turned off ***

You can find the poster of the event in the attachment.
We are looking forward to seeing you!

For further information, please visit our website:
https://sites.google.com/view/insalate-di-matematica or contact us at
insalate.matematica a unimib.it. Find us also on our Instagram page:
https://www.instagram.com/insalate_di_matematica23/
<https://www.instagram.com/insalate_di_matematica22/>.

The organizers: Andrea Bisterzo, Simone Blumer, Alberto Cassella, Bianca
Marchionna, Andrea Rivezzi, Giovanni Siclari, Marta Tameni, Matteo
Tarocchi, Marco Zullino.

Il giorno mer 10 mag 2023 alle ore 10:00 "Insalate Di Matematica" <
insalate.matematica a unimib.it> ha scritto:

> Dear all,
>
> As part of the series of seminars "Insalate di Matematica",*  Elisabetta
> Masut* (Università degli Studi di Padova) will give a talk.
> The speaker will deliver the talk *in presence* and the meeting will also
> be broadcasted *online* (see below for more information).
>
> Here the details:
>
> *Date and time*: *17th of May 2023, 4:00 pm (CET)*
>
> *Title*: "Non-existence of integral Hopf orders for twists of
> several simple groups of Lie type."
>
> *Abstract*:
> In 1975 Kaplansky listed 10 conjectures on Hopf algebras, which have been
> the focus of a great deal of research. Some of these conjectures are still
> unanswered. In particular, we will be interested on the sixth one, that is
> a generalization of Frobenius theorem in the framework of representation
> theory for Hopf algebras.
> It states that, given a complex finite-dimensional semisimple Hopf algebra [image:
> H], the dimension of every irreducible representation of [image: H] divides
> the dimension of [image: H].
> Larson proved a weaker version of it: if the complex finite-dimensional
> semisimple Hopf algebra admits a Hopf order over a number ring, then
> Kaplansky' sixth conjecture is satisfied.
> A natural question now arises: does every complex
> semisimple finite-dimensional Hopf algebra which satisfies Kaplansky'
> sixth conjecture admit a Hopf order over a number ring?
> The answer is negative.
> In this talk, after a brief excursus on Hopf algebras, we present families
> of Hopf algebras, which satisfy Kamplansky' sixth conjecture, but they do
> not admit a Hopf order over a number ring. These Hopf algebras are
> constructed as Drinfel'd twist of the group algebras [image: KG], with [image:
> K] a number field. We will prove that for every finite simple group [image:
> G], there is always a deformation, such that the twisted Hopf algebra
> does not admit a Hopf order over a number ring. Moreover, we will show that
> for two families of groups, this non-existence result holds for any twist.
> This talk is based on a joint work with Giovanna Carnovale and Juan Cuadra.
>
> *Keywords*: Hopf algebras, Hopf orders, Drinfel'd twist
>
> *Information to attend in room 3014*
> The seminar will take place in room 3014, at the building U5-Ratio,
> Università degli Studi di Milano Bicocca.
>
> *Information to attend online *
>
> https://unimib.webex.com/unimib/j.php?MTID=ma9fac62549ae03a1196907bfff0a7057 (Password:
> Insalate (46725283 from phones))
>
> *** We inform you that the talk will be recorded and uploaded on our
> website. If you join the talk after the starting time, we kindly ask you to
> ensure that your microphone and webcam are turned off ***
>
> You can find the poster of the event in the attachment.
> We are looking forward to seeing you!
>
> For further information, please visit our website:
> https://sites.google.com/view/insalate-di-matematica or contact us at
> insalate.matematica a unimib.it. Find us also on our Instagram page:
> https://www.instagram.com/insalate_di_matematica23/
> <https://www.instagram.com/insalate_di_matematica22/> .
>
> The organizers: Andrea Bisterzo, Alberto Cassella, Bianca Marchionna,
> Andrea Rivezzi, Simone Blumer, Giovanni Siclari, Marta Tameni, Matteo
> Tarocchi, Marco Zullino.
>
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