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.
A root datum consists of a quadruple where The elements of
, then the root datum is called reduced.
with a split maximal torus
then its root datum is a quadruple where A connected split reductive algebraic group over
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.
, we can define a dual root datum
is the complex connected reductive group whose root datum is dual to that of