For questions involving abstract root systems, their associated Weyl groups and Dynkin diagrams, as well as their applications to Lie theory, graph theory, or other related fields.
Given a finite-dimensional Euclidean vector space V, a root system in V is a subset of nonzero vectors satisfying some fairly restrictive axioms. As such, in any given dimension there are only finitely many possible isomorphism classes of root systems.
Irreducible root systems are those which cannot be 'built' from smaller root systems in a definable way. Irreducible root systems are completely classified by their associated connected Dynkin diagrams, with four infinite families (called the classical root systems) and 5 exceptional cases.
Weyl groups are subgroups of the isometry group of root systems.
Cartan subalgebras (h) of semisimple Lie algebras (g) have a root system ($\Phi$) relation, as indicated by $\dim g = \dim h+ |\Phi|$.
