Algebra Seminar: On algebraic supergroups and their representations

Monday, January 13 at 3:30pm to 4:30pm

This is a past event.

Kaprielian Hall (KAP), 245
3620 South Vermont Avenue, Los Angeles, CA 90089

Taiki Shibata, Okayama University of Science

Abstract:
An algebraic supergroup is a group-valued functor on the category of
commutative superalgebras ($=Z/2Z$-graded algebra) represented by a finitely
generated commutative Hopf superalgebra. It has been known that
representations of algebraic supergroups can be applied in non-super
(modular) representation theory. In 2011, V. Serganova introduced the notion
of quasireductive supergroups as a super-version of the notion of split
reductive groups. They form a large class of algebraic supergroups including
Chevalley supergroups (introduced by R. Fioresi and F. Gavarini, 2012) and
queer supergroups $Q(n)$ (whose Lie superalgebra is a queer superalgebra
$q(n)$). She constructed irreducible representations of quasireductive
supergroups over an algebraically closed field of characteristic zero, in
terms of their Lie superalgebras.

In this talk, I will explain a Hopf-algebraic approach to the study of
quasireductive supergroups and, as an application, I will give a
generalization of Serganova's construction to the case when the base field
is arbitrary. The main tool of the construction is a super-version of the
hyperalgebra $hy(G)$ of an algebraic group $G$ (due to M. Takeuchi) which is
a refinement of the notion of the Lie algebra $Lie(G)$ of $G$.

Event Type

Lecture / Talk / Workshop

Campus

University Park Campus

Department
Mathematics
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