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.

A root datum consists of a quadruple

where

The elements of ${\displaystyle \Phi }$ are called the roots of the root datum, and the elements of ${\displaystyle \Phi ^{\vee }}$ are called the coroots.

If ${\displaystyle \Phi }$ does not contain ${\displaystyle 2\alpha }$ for any ${\displaystyle \alpha \in \Phi }$, then the root datum is called reduced.

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