A root datum records both the root system of a connected reductive algebraic group and its integral character lattice of an algebraic torus. It consists of finite-rank free abelian groups with a perfect bilinear map , finite sets and , and a bijection satisfying . The associated reflectionspermute the roots and coroots, respectively, compatibly with the correspondence: . They generate the Weyl group.
For a split maximal algebraic torus in a reductive algebraic group, and , paired by composition. For , using gives , with root and coroot . For the root is primitive and the coroot is twice a generator. These groups have the same A1 root system but different root data: the integral lattices retain the information lost by passing to the Lie algebra.
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In the context of algebraic groups and Lie algebras, a **root datum** is a structured way of encoding certain aspects of the symmetries and properties of these mathematical objects. Specifically, a root datum consists of the following components: 1. **A finite set of roots**: These are usually vectors in a Euclidean space, which can be thought of as directions that reflect the symmetries of the system.