Lie Superalgebra Day at Ben Gurion University
December 25th, 2018
Ben Gurion University, Marcus campus, Beer Sheva
Deichmann Bldg. (58), Room -101 (map).
10:30-11:30 Shifra Reif (Bar Ilan University),
An introduction to Lie superalgebras, the classical and the strange.
Weyl's theorem asserts that representations of simple Lie algebras admit complete reducibility over a field of characteristic zero. The characters of finite-dimensional modules is given by the Weyl character formula, and the Grothendieck ring is isomorphic to ring of symmetric Laurent polynomials. For Lie superalgebras, there is no complete reducibility and the problem of obtaining a character formula is a main subject of research for more than two decades.
In the talk, we shall introduce Lie superalgebras via two examples: the general linear Lie superalgebras gl(m|n) which belongs to the basic-classical series and the periplectic Lie superalgebra p(n) which belongs to the strange series. As far as time permits, we describe methods used to overcome the above difficulties: translation functors, Duflo--Serganova functor and their effect on the Grothendieck ring.
11:35-12:35 Crystal Hoyt (Ort Braude),
Representations of sl(infinity) and category O for gl(m|n).
The Lie algebra sl(infinity) is a direct limit of the Lie algebras sl(n) as n goes to infinity. Brundan, Losev and Webster showed that the Lie algebra sl(infinity) acts by endofunctors on a certain category of modules for the Lie superalgebra gl(m|n), where the endofunctors are so-called translation functors.
We will define the (reduced) complexified Grothendieck group of this category, which is a complex vector space, and explain that it inherits a natural sl(infinity)-action. We will then describe the structure of this sl(infinity)-module.
This is a joint work with I. Penkov, V. Serganova.
14:30-15:30 Vera Serganova (UC Berkeley),
Borel-Weil-Bott theorem for algebraic supergroups and weak BGG reciprocity.
We will review some results about superanalogue of Borel-Weil-Bott theorem, explain the role of Weyl groupoid and prove a weak version of BGG reciprocity.
Then we illustrate how BGG reciprocity can be used for computing the Cartan matrix of the category of finite dimensional representations of the nontivial central extension of the periplectic supergroup P(4).
16:00-17:00 Kevin Coulembier (University of Sydney),
Ringel duals of Brauer algebras via supergroups.
The Brauer algebra describes the invariant theory of orthogonal and symplectic groups. It is also an interesting example of a cellular or quasi-hereditary algebra. A more recent interpretation sees the Brauer algebras as the endomorphism algebras in a universal monoidal category.
The family of quasi-hereditary algebras is equipped with a duality discovered by Ringel. I will demonstrate that we can realise the Ringel dual of the Brauer algebra as an abelian subcategory of a category of algebraic representations of an affine supergroup scheme. This is inspired and motivated by the construction of universal abelian monoidal categories using supergroups by Entova, Hinich and Serganova.
Organizer: Inna Entova-Aizenbud, email@example.com.
This event is supported by the Center for Advanced Studies in Mathematics at Ben Gurion University, and by the Faculty of Natural Sciences Distinguished Scientist Visitors Program at Ben Gurion University.
If you are interested in attending, please register here (no fee required).