Generalised Root Systems in Algebra, Geometry, and Mathematical Physics

Abstract

"Nachdiplomvorlesung"

Content

Root systems of simple complex Lie algebras and their Dynkin diagrams are amongst the most remarkable objects in mathematics, appearing in various problems from algebra to singularity theory. A related geometric theory of regular polyhedra and Coxeter groups has a long history going back to ancient time. In the 1990s some intriguing deformations of roots systems were discovered in the theory of quantum Calogero-Moser systems. Further investigations have revealed deep links of these deformations with Lie superalgebras and some classical problems of algebra and mathematical physics. The aim of the course is to introduce and to study three interrelated notions, which can be considered as generalisations of root systems. Topics to be covered include the following. 1) Locus configurations of hyperplanes, monodromy-free Schrödinger operators and the Hadamard problem in the theory of Huygens principle. 2) Generalised root systems in the sense of Serganova, simple Lie superalgebras and deformed quantum Calogero-Moser systems. Relation with Cayley-Sylvester coincident root loci problem: how to determine when a polynomial has multiple roots of given multiplicity? 3) V-systems, logarithmic Frobenius structures and Coxeter discriminants. Relation with the theory of Witten-Dijkgraaf-Verlinde-Verlinde equation. A special feature of the course is that there will be many open problems reflecting the current stage of the subject.

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