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SUMMARY:Vera Serganova (University of California\, Berkeley)
DTSTART;VALUE=DATE-TIME:20200703T183000Z
DTEND;VALUE=DATE-TIME:20200703T192000Z
DTSTAMP;VALUE=DATE-TIME:20240329T132713Z
UID:T-Rep/1
DESCRIPTION:Title: Th
e Jacobson-Morozov theorem for Lie superalgebras via semisimplification fu
nctor for tensor categories\nby Vera Serganova (University of Californ
ia\, Berkeley) as part of T-Rep: A midsummer night's session on representa
tion theory and tensor categories\n\n\nAbstract\nThe celebrated Jacobson-M
orozov theorem claims that every nilpotent element of a semisimple Lie alg
ebra g can be embedded into an sl(2)-triple inside g. Let g be a Lie super
algebra with reductive even part and x be an odd element of g with non-zer
o nilpotent [x\,x]. We give necessary and sufficient condition when x can
be embedded in osp(1|2) inside g. The proof follows the approach of Etingo
f and Ostrik and involves semisimplification functor for tensor categories
. Next\, we will show that for every odd x in g we can construct a symmetr
ic monoidal functor between categories of representations of certain super
algebras. We discuss some properties of these functors and applications of
them to representation theory of superalgebras with reductive even part.
We also discuss possible generalization of reductive envelope of an algebr
aic group to the case of a supergroup. (Joint work with Inna Entova-Aizenb
ud).\n
LOCATION:https://researchseminars.org/talk/T-Rep/1/
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BEGIN:VEVENT
SUMMARY:Pavel Etingof (MIT)
DTSTART;VALUE=DATE-TIME:20200703T201000Z
DTEND;VALUE=DATE-TIME:20200703T210000Z
DTSTAMP;VALUE=DATE-TIME:20240329T132713Z
UID:T-Rep/2
DESCRIPTION:Title: Ne
w incompressible symmetric tensor categories in positive characteristic\nby Pavel Etingof (MIT) as part of T-Rep: A midsummer night's session on
representation theory and tensor categories\n\n\nAbstract\nLet k be an al
gebraically closed field of characteristic p>0. The category of tilting mo
dules for SL2(k) has a tensor ideal In generated by the n-th Steinberg mod
ule. I will explain that the quotient of the tilting category by In admits
an abelian envelope\, a finite symmetric tensor category Verpn\, which is
not semisimple for n>1. This is a reduction to characteristic p of the se
misimplification of the category of tilting modules for the quantum group
at a root of unity of order pn. These categories are incompressible\, i.e.
do not admit fiber functors to smaller categories. For p=1\, these catego
ries were defined by S. Gelfand and D. Kazhdan and by G. Georgiev and O. M
athieu in early 1990s\, but for n>1 they are new. I will describe these ca
tegories in detail and explain a conjectural formulation of Deligne's theo
rem in characteristic p in which they appear. This is joint work with D. B
enson and V. Ostrik.\n
LOCATION:https://researchseminars.org/talk/T-Rep/2/
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BEGIN:VEVENT
SUMMARY:Kevin Coulembier (University of Sydney)
DTSTART;VALUE=DATE-TIME:20200703T211000Z
DTEND;VALUE=DATE-TIME:20200703T220000Z
DTSTAMP;VALUE=DATE-TIME:20240329T132713Z
UID:T-Rep/3
DESCRIPTION:Title: Mo
noidal abelian envelopes\nby Kevin Coulembier (University of Sydney) a
s part of T-Rep: A midsummer night's session on representation theory and
tensor categories\n\n\nAbstract\nThe notion of an abelian envelope of a k-
linear rigid monoidal category emerged rather recently from the constructi
ons by Entova-Hinich-Serganova and Comes-Ostrik of universal tensor catego
ries. They subsequently reappeared in a construction by Benson-Etingof of
an intriguing family of incompressible tensor categories in characteristic
2. All of this clearly demanded a thorough exploration of this concept of
abelian envelopes. In this talk I will report on recent progress on the m
atter due to a number of authors.\n
LOCATION:https://researchseminars.org/talk/T-Rep/3/
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