  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 for short). 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.
