Given some 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 diagram, with four infinite families (called the classical root systems) and 5 exceptional cases. Root systems are a key component to the classification of simple Lie groups and simple Lie algebras over algebraically closed fields.
This tag welcomes any and all question related to abstract root systems and their association to algebraic/combinatoric structures such as Weyl groups, lie groups, lie algebras, and Dynkin diagrams.