Generalized root systems

Abstract

We generalize the notion of a root system by relaxing the conditions that ensure that it is invariant under reflections and study the resulting structures, which we call generalized root systems (GRSs). Since both Kostant root systems and root systems of Lie superalgebras are examples of GRSs, studying GRSs provides a uniform axiomatic approach to studying both of them. GRSs inherit many of the properties of root systems. In particular, every GRS defines a crystallographic hyperplane arrangement. We believe that GRSs provide an intrinsic counterpart to finite Weyl groupoids and crystallographic hyperplane arrangements, extending the relationship between finite Weyl groupoids and crystallographic hyperplane arrangements established by Cuntz. An important difference between GRSs and root systems is that GRSs may lack a (large enough) Weyl group. In order to compensate for this, we introduce the notion of a virtual reflection, building on a construction of Penkov and Serganova in the context of root systems of Lie superalgebras. The most significant new feature of GRSs is that, along with subsystems, one can define quotient GRSs. Both Kostant root systems and root systems of Lie superalgebras are equivalent to quotients of root systems and all root systems are isomorphic to quotients of simply-laced root systems. We classify all rank 2 GRSs and show that they are equivalent to quotients of root systems. Finally, we discuss in detail quotients of root systems. In particular we provide all isomorphisms and equivalences among them. Our results on quotient of root systems provide a different point of view on flag manifolds, reproving results of Alekseevsky and Graev.