Root datum

In mathematical group theory, the root datum of a connected split reductive algebraic group over a field is a generalization of a root system that determines the group up to isomorphism. They were introduced by Michel Demazure in Grothendieck's Séminaire [de géométrie algébrique|SGA III], published in 1970.

Definition

A root datum consists of a quadruple
where

and are free abelian groups of finite rank together with a perfect pairing between them with values in which we denote by .
is a finite subset of and is a finite subset of and there is a bijection from onto, denoted by.
For each,.
For each, the map induces an automorphism of the root datum

The elements of are called the roots of the root datum, and the elements of are called the coroots.
If does not contain for any, then the root datum is called reduced.

The root datum of an algebraic group

If is a reductive algebraic group over an algebraically closed field with a split maximal torus then its root datum is a quadruple
where

is the lattice of characters of the maximal torus,
is the dual lattice,
is a set of roots,
is the corresponding set of coroots.

A connected split reductive algebraic group over is uniquely determined by its root datum, which is always reduced. Conversely for any root datum there is a reductive algebraic group. A root datum contains slightly more information than the Dynkin diagram, because it also determines the center of the group.
For any root datum, we can define a dual root datum by switching the characters with the 1-parameter subgroups, and switching the roots with the coroots.
If is a connected reductive algebraic group over the algebraically closed field, then its Langlands dual group is the complex connected reductive group whose root datum is dual to that of.