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 SGA III, published in 1970.
Definition.
A root datum consists of a quadruple
where 
The elements of formula_5 are called the roots of the root datum, and the elements of formula_7 are called the coroots.
If formula_5 does not contain formula_24 for any formula_25, then the root datum is called reduced.
The root datum of an algebraic group.
If formula_26 is a reductive algebraic group over an algebraically closed field formula_27 with a split maximal torus formula_28 then its root datum is a quadruple 
where 
A connected split reductive algebraic group over formula_27 is uniquely determined (up to isomorphism) 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 formula_29, we can define a dual root datum formula_36 by switching the characters with the 1-parameter subgroups, and switching the roots with the coroots.
If formula_26 is a connected reductive algebraic group over the algebraically closed field formula_27, then its Langlands dual group formula_39 is the complex connected reductive group whose root datum is dual to that of formula_26.