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==The root datum of an algebraic group==
If ''<math>G''</math> is a reductive algebraic group over an [[algebraically closed field]] ''<math>K''</math> with a split maximal torus ''<math>T''</math> then its '''root datum''' is a quadruple
:<math>(''X''<sup>^*</sup>, &\Phi;, ''X''<sub>X_*</sub>, &\Phi;<sup>v^{\vee})</supmath>),
where
*''X''<supmath>X^*</supmath> is the lattice of characters of the maximal torus,
*''X''<submath>X_*</submath> is the dual lattice (given by the 1-parameter subgroups),
*&<math>\Phi;</math> is a set of roots,
*Φ<supmath>v\Phi^{\vee}</supmath> is the corresponding set of coroots.
A connected split reductive algebraic group over ''<math>K''</math> 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 <math>(''X''<sup>^*</sup>, &\Phi;,''X''<sub> X_*</sub>, &\Phi;<sup>v^{\vee})</supmath>), we can define a '''dual root datum''' (''X''<submath>(X_*</sub>, &\Phi;<sup>v</sup>^{\vee},''X''<sup>^*</sup>, &\Phi;)</math> by switching the characters with the 1-parameter subgroups, and switching the roots with the coroots.
If ''<math>G''</math> is a connected reductive algebraic group over the algebraically closed field ''<math>K''</math>, then its [[Langlands dual group]] <supmath>''{}^L'' G</supmath>''G'' is the complex connected reductive group whose root datum is dual to that of ''<math>G''</math>.
==References==
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