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
 * $$(X^\ast, \Phi, X_\ast, \Phi^\vee)$$,

where
 * $$X^\ast$$ and $$X_\ast$$ are free abelian groups of finite rank together with a perfect pairing between them with values in $$\mathbb{Z}$$ which we denote by (in other words, each is identified with the dual of the other).
 * $$\Phi$$ is a finite subset of $$X^\ast$$ and $$\Phi^\vee$$ is a finite subset of $$X_\ast$$ and there is a bijection from $$\Phi$$ onto $$\Phi^\vee$$, denoted by $$\alpha\mapsto\alpha^\vee$$.
 * For each $$\alpha$$, $$(\alpha, \alpha^\vee)=2$$.
 * For each $$\alpha$$, the map $$x\mapsto x-(x,\alpha^\vee)\alpha$$ induces an automorphism of the root datum (in other words it maps $$\Phi$$ to $$\Phi$$ and the induced action on $$X_\ast$$ maps $$\Phi^\vee$$ to $$\Phi^\vee$$)

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 $$\alpha\in\Phi$$, 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
 * $$(X^*, \Phi, X_*, \Phi^{\vee})$$,

where
 * $$X^*$$ is the lattice of characters of the maximal torus,
 * $$X_*$$ is the dual lattice (given by the 1-parameter subgroups),
 * $$\Phi$$ is a set of roots,
 * $$\Phi^{\vee}$$ is the corresponding set of coroots.

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 $$(X^*, \Phi, X_*, \Phi^{\vee})$$, we can define a dual root datum $$(X_*, \Phi^{\vee},X^*, \Phi)$$ 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 $$G$$.