Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

DMV Tagung 2011 - Köln, 19. - 22. September
Shun-Jen Cheng
Academia Sinica, Taiwan
Kostant homology formula for oscillator representations of classical Lie superalgebras
Pierluigi Möseneder Frajria
Politecnico di Milano
Superalgebras and Theta correspondence over the real numbers
The odd part of the super denominator of a basic classical Lie superalgebra is the character of the
oscillator representation of the odd part of the superalgebra. It is then natural to use superalgebra
techniques to study this representation. In particular, we will show how the generalized denominator
formulas for basic classical Lie superalgebras can be used to derive the Theta correspondence between
representations of a compact dual pair.
Fabio Gavarini
Università degli Studi di Roma “Tor Vergata”
Algebraic supergroups associated to simple Lie superalgebras
For any finite dimensional (complex) simple Lie superalgebra I provide an explicit recipe to construct an
algebraic supergroupe G (defined via its functor of points) whose tangent Lie superalgebra is g itself.To
do that, I generalise the classical Chevalley method, which constructs a (semi-) simple algebraic group
starting from any complex, f. d. (semi-) simple Lie algebra g and from a faithful f. d. g-module V: the basic
ingredient to make use of is the datum of a so-called “Chevalley basis”. I shall show that one can do
the same when g is replaced with a simple Lie superalgebra: one introduces then a notion of “Chevalley
basis”, one proves the existence of the latter, and then one essentially implements the same method (with
many new aspects to deal with, of course). A remarkable fact is that this strategy is succesful both with
the (simple) Lie superalgebras of classical type and with those of Cartan type—somehow extending the
range of application of Chevalley’s original idea.
Besides this “existence” result, I shall present also a “uniqueness” one: every connected algebraic supergroup
whose Lie superalgebra be (f.d.) simple is isomorphic to one of the supergroups that I just
constructed. Thus one eventually finds a complete classification of such supergroups.
Literatur
Fioresi, R., Gavarini, F. (2011). Chevalley Supergroups. Mem. Amer. Math. Soc., in press. ar-
Xiv:0808.0785.
Gavarini, F. (2008). Chevalley Supergroups of type D(2,1;a). arXiv:1008.1838.
Fioresi, R., Gavarini, F. (2010). On the construction of Chevalley Supergroups. In: Supersymmetry in
Mathematics and Physics, Lect. Notes Math., to appear. arXiv:1008.1838.
Fioresi, R., Gavarini, F. Algebraic supergroups with classical Lie superalgebras, in preparation.
Gavarini, F. Algebraic supergroups of Cartan type, in preparation.
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