Research Seminar Algebra, Number Theory and Discrete Mathematics

Frieze patterns and simplicial arrangements

Arrangements of lines which triangulate the plane, the so-called simplicial arrangements, appear to be rare. They have been collected during the last 80 years. It is conjectured that they are all known. In my talk I will explain a relation between simplicial arrangements and frieze patterns.

Pseudo Root Systems and Simplicial Hyperplane Arrangements

Root systems are sets of vectors, which elegantly describe all finite reflection groups. Dimitrov and Fioresi discovered generalized root system, whose short definition generalizes the concept of the classical root systems. Indeed the corresponding hyperplane arrangements are still simplicial hyperplane arrangements and Cuntz and Mühlherr classified them to be the Weyl arrangements and their restrictions. We generalize the Dimitrov and Fioresis root systems to pseudo root systems. The hyperplane arrangements of pseudo root systems are still simplicial hyperplane arrangements. We prove that pseudo root system are closed under restriction and localization and all reflection arrangements and their restrictions have pseudo root systems.Furthermore, we show that there are only finitely many 3-dimensional pseudo root systems and more simplicial hyperplane arrangements with pseudo root systems then restrictions of reflection arrangements.
 

Weyl Groupoids for Lie Superalgebras

For Lie superalgebras the Weyl group is not as powerful as for Lie algebras, and in particular not all Borel subalgebras are conjugate under the Weyl group action. To fix this it has been proposed that the Weyl group should be replaced by a "Weyl groupoid", which also includes the odd reflections. In my talk I will explain how the notion of Weyl groupoid introduced by Heckenberger in the theory of Nichols algebras 
can be applied to contragredient Lie superalgebras following Heckenberger, Schneider and Yamane, and I will provide an explicit combinatorial description of the Weyl groupoids of sl(m|n) and osp(r|2n). I will also discuss the relation to Serganova's definition of Weyl groupoid and the root groupoid  introduced by Gorelik, Hinich and Serganova, and potential applications to representation theory. The talk is based on joint work with Jonas Nehme.