<div dir="ltr"><font color="#000000" face="arial, sans-serif">Dear all,</font><div><font color="#000000" face="arial, sans-serif"><br></font><div dir="auto"><font color="#000000" face="arial, sans-serif">We remind you the upcoming talk of "Insalate di Matematica".</font></div><div dir="auto"><font color="#000000" face="arial, sans-serif">Here the details:</font></div><div dir="auto"><font color="#000000" face="arial, sans-serif"><br></font></div><div dir="auto"><font color="#000000"><font face="arial, sans-serif"><i><u>Speaker</u>:</i> </font><b>Elisabetta Masut</b><span class="gmail-Apple-converted-space"> </span>(Università degli Studi di Padova)<span class="gmail-Apple-converted-space"> </span></font></div><span class="gmail-im"><font face="arial, sans-serif"><font color="#000000"><div dir="auto"><br></div><div dir="auto"><i><u>Date and time</u>:</i> <b style="font-family:Arial,Helvetica,sans-serif">17th of May 2023, 4:15 pm (CET)</b></div></font></font><font face="arial, sans-serif"><font color="#000000"><div dir="auto"><br></div></font></font></span><div dir="auto"><u><i><font color="#000000" face="arial, sans-serif">Information to attend</font></i></u></div><div dir="auto"><font color="#000000" face="arial, sans-serif"><i><u>In presence</u>:</i> The seminar will take place in room 3014, at the building U5-Ratio, Università degli Studi di Milano Bicocca.</font></div><div dir="auto"><font color="#000000"><font face="arial, sans-serif"><u><i>Online</i></u><i>:</i> The seminar will be streamed via Webex platform at the following link: </font><a href="https://unimib.webex.com/unimib/j.php?MTID=ma9fac62549ae03a1196907bfff0a7057" target="_blank">https://unimib.webex.com/unimib/j.php?MTID=ma9fac62549ae03a1196907bfff0a7057</a><span class="gmail-Apple-converted-space"> </span>(Password: Insalate (46725283 from phones))</font></div><font color="#000000"><font face="arial, sans-serif"><span class="gmail-im"><div><b><i><br></i></b></div><div dir="auto"><u><i>Title</i></u><i>:</i> <span style="font-family:Arial,Helvetica,sans-serif;background-color:rgb(255,255,255)">"Non-existence of integral Hopf orders for twists of several simple groups of Lie type."</span></div></span></font><font face="arial, sans-serif"><span class="gmail-im"><div dir="auto"><br></div><div dir="auto"><u><i>Abstract</i></u>:</div></span></font>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.<br>It states that, given a complex finite-dimensional semisimple Hopf algebra<span class="gmail-Apple-converted-space"> </span><img alt="H" title="H" src="https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=H" id="m_-4809126886221012312m_7273706922260474265m_-1944475522459475501m_58720117783386291l0.7405293183421193" height="11" width="13" class="gmail-CToWUd" style="display: inline;">, the dimension of every irreducible representation of<span class="gmail-Apple-converted-space"> </span><img alt="H" title="H" src="https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=H" id="m_-4809126886221012312m_7273706922260474265m_-1944475522459475501m_58720117783386291l0.4853673310578437" height="11" width="13" class="gmail-CToWUd" style="display: inline;"> divides the dimension of<span class="gmail-Apple-converted-space"> </span><img alt="H" title="H" src="https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=H" id="m_-4809126886221012312m_7273706922260474265m_-1944475522459475501m_58720117783386291l0.7771657738602054" height="11" width="13" class="gmail-CToWUd" style="display: inline;">. <br>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. <br>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?<br>The answer is negative. <br>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 <img alt="KG" title="KG" src="https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=KG" id="m_-4809126886221012312m_7273706922260474265m_-1944475522459475501m_58720117783386291l0.7935715250541977" height="11" width="26" class="gmail-CToWUd" style="display: inline; vertical-align: -0.333px;">, with<span class="gmail-Apple-converted-space"> </span><img alt="K" title="K" src="https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=K" id="m_-4809126886221012312m_7273706922260474265m_-1944475522459475501m_58720117783386291l0.2687113625462745" height="11" width="13" class="gmail-CToWUd" style="display: inline;"><span class="gmail-Apple-converted-space"> </span>a number field. We will prove that for every finite simple group<span class="gmail-Apple-converted-space"> </span><img alt="G" title="G" src="https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=G" id="m_-4809126886221012312m_7273706922260474265m_-1944475522459475501m_58720117783386291l0.026336257026702725" height="11" width="11" class="gmail-CToWUd" style="display: inline; vertical-align: -0.333px;">, 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.<br>This talk is based on a joint work with Giovanna Carnovale and Juan Cuadra.<font face="arial, sans-serif"><span class="gmail-im"><div dir="auto"><br></div><div dir="auto"><i><u>Keywords</u>: </i><span style="font-family:Arial,Helvetica,sans-serif">Hopf algebras, Hopf orders, Drinfel'd twist.</span></div></span></font><font face="arial, sans-serif"><div dir="auto"><br></div><span class="gmail-im"><div dir="auto"><b>** 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 **</b><br></div><div dir="auto"><br></div><div dir="auto">You can find the poster of the event in the attachment. </div><div dir="auto">We are looking forward to seeing you! <br></div><div dir="auto"><br></div></span></font></font><div dir="auto"><font face="arial, sans-serif" color="#000000">For further information, please visit our website: <a href="https://sites.google.com/view/insalate-di-matematica" rel="noreferrer noreferrer noreferrer noreferrer noreferrer noreferrer noreferrer noreferrer" target="_blank">https://sites.google.com/view/insalate-di-matematica</a> or contact us at <a href="mailto:insalate.matematica@unimib.it" rel="noreferrer noreferrer noreferrer noreferrer noreferrer noreferrer noreferrer noreferrer" target="_blank">insalate.matematica@unimib.it</a>. Find us also on our Instagram page: <a href="https://www.instagram.com/insalate_di_matematica22/" rel="noreferrer noreferrer noreferrer noreferrer noreferrer noreferrer noreferrer noreferrer" target="_blank">https://www.instagram.com/insalate_di_matematica23/</a>.</font></div><div dir="auto"><font color="#000000" face="arial, sans-serif"><br></font></div><div dir="auto"><font color="#000000" face="arial, sans-serif">The organizers: Andrea Bisterzo, Simone Blumer, Alberto Cassella, Bianca Marchionna, Andrea Rivezzi, Giovanni Siclari, Marta Tameni, Matteo Tarocchi, Marco Zullino.</font></div></div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">Il giorno mer 10 mag 2023 alle ore 10:00 "Insalate Di Matematica" <<a href="mailto:insalate.matematica@unimib.it">insalate.matematica@unimib.it</a>> ha scritto:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left-width:1px;border-left-style:solid;border-left-color:rgb(204,204,204);padding-left:1ex"><div dir="ltr"><div dir="ltr">Dear all,<div dir="auto"><br></div><div dir="auto">As part of the series of seminars "Insalate di Matematica",<b> Elisabetta Masut</b> (Università degli Studi di Padova) will give a talk. </div><div dir="auto">The speaker will deliver the talk <u>in presence</u> and the meeting will also be broadcasted <u>online</u> (see below for more information).</div><div dir="auto"><br></div><div dir="auto">Here the details:</div><div dir="auto"><br></div><div dir="auto"><u>Date and time</u>: <b>17th of May 2023, 4:00 pm (CET)</b></div><div dir="auto"><br></div><div dir="auto"><u>Title</u>: <span style="background-color:rgb(255,229,153)">"Non-existence of integral Hopf orders for twists of several simple groups of Lie type."</span></div><div dir="auto"><br></div><div dir="auto"><u>Abstract</u>:</div>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.<br>It states that, given a complex finite-dimensional semisimple Hopf algebra <img alt="H" title="H" src="https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=H" id="m_-4809126886221012312m_7273706922260474265m_-1944475522459475501m_58720117783386291l0.7405293183421193" style="display: inline;" height="11" width="13">, the dimension of every irreducible representation of <img alt="H" title="H" src="https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=H" id="m_-4809126886221012312m_7273706922260474265m_-1944475522459475501m_58720117783386291l0.4853673310578437" style="display: inline;" height="11" width="13"> divides the dimension of <img alt="H" title="H" src="https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=H" id="m_-4809126886221012312m_7273706922260474265m_-1944475522459475501m_58720117783386291l0.7771657738602054" style="display: inline;" height="11" width="13">. <br>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. <br>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?<br>The answer is negative. <br>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 <img alt="KG" title="KG" src="https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=KG" id="m_-4809126886221012312m_7273706922260474265m_-1944475522459475501m_58720117783386291l0.7935715250541977" style="display: inline; vertical-align: -0.333px;" height="11" width="26">, with <img alt="K" title="K" src="https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=K" id="m_-4809126886221012312m_7273706922260474265m_-1944475522459475501m_58720117783386291l0.2687113625462745" style="display: inline;" height="11" width="13"> a number field. We will prove that for every finite simple group <img alt="G" title="G" src="https://s0.wp.com/latex.php?zoom=3&bg=ffffff&fg=000000&s=0&latex=G" id="m_-4809126886221012312m_7273706922260474265m_-1944475522459475501m_58720117783386291l0.026336257026702725" style="display: inline; vertical-align: -0.333px;" height="11" width="11">, 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.<br>This talk is based on a joint work with Giovanna Carnovale and Juan Cuadra.<div dir="auto"><br></div></div><div dir="ltr"><div dir="auto"><u>Keywords</u>: Hopf algebras, Hopf orders, Drinfel'd twist</div><div dir="auto"><br></div><div dir="auto"></div><div dir="auto"><u><b>Information to attend in room 3014</b></u></div><div dir="auto">The seminar will take place in room 3014, at the building U5-Ratio, Università degli Studi di Milano Bicocca.</div><div dir="auto"><br></div><div dir="auto"></div><div dir="auto"><u><b>Information to attend online </b></u></div><div dir="auto"><font color="#0000ff"><a href="https://unimib.webex.com/unimib/j.php?MTID=ma9fac62549ae03a1196907bfff0a7057" target="_blank">https://unimib.webex.com/unimib/j.php?MTID=ma9fac62549ae03a1196907bfff0a7057</a></font><font color="#00ffff"> </font>(Password: Insalate (46725283 from phones))</div><div dir="auto"><br></div><div dir="auto"></div><div dir="auto"><b>** 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 **</b></div><div dir="auto"><b><br></b></div><div dir="auto"></div><div dir="auto">You can find the poster of the event in the attachment. </div><div dir="auto">We are looking forward to seeing you! </div><div dir="auto"><br></div><div dir="auto"></div><div dir="auto">For further information, please visit our website: <a href="https://sites.google.com/view/insalate-di-matematica" rel="noreferrer noreferrer noreferrer noreferrer noreferrer noreferrer noreferrer noreferrer noreferrer noreferrer" target="_blank">https://sites.google.com/view/insalate-di-matematica</a> or contact us at <a href="mailto:insalate.matematica@unimib.it" rel="noreferrer noreferrer noreferrer noreferrer noreferrer noreferrer noreferrer noreferrer noreferrer noreferrer" target="_blank">insalate.matematica@unimib.it</a>. Find us also on our Instagram page: <a href="https://www.instagram.com/insalate_di_matematica22/" rel="noreferrer noreferrer noreferrer noreferrer noreferrer noreferrer noreferrer noreferrer noreferrer noreferrer" target="_blank">https://www.instagram.com/insalate_di_matematica23/</a> .</div><div dir="auto"><br></div><div dir="auto"></div><div dir="auto">The organizers: Andrea Bisterzo, Alberto Cassella, Bianca Marchionna, Andrea Rivezzi, <span style="color:rgb(0,0,0)">Simone Blumer, </span>Giovanni Siclari, Marta Tameni, Matteo Tarocchi, Marco Zullino.</div></div></div>
</blockquote></div>