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 \Phi are called the roots of the root datum, and the elements of \Phi^\vee are called the coroots. If \Phi does not contain 2\alpha for any, then the root datum is called reduced.

The root datum of an algebraic group

If G is a reductive algebraic group over an algebraically closed field K with a split maximal torus T then its root datum is a quadruple where A connected split reductive algebraic group over K 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, we can define a dual root datum by switching the characters with the 1-parameter subgroups, and switching the roots with the coroots. If G is a connected reductive algebraic group over the algebraically closed field K, then its Langlands dual group {}^L G is the complex connected reductive group whose root datum is dual to that of G.