 Root datum

A root datum is a quadruple $(X^{\ast },\Phi ,X_{\ast },{\Phi }^{\vee })$ of dual lattices, roots, and coroots with the natural perfect pairing and reflection axioms. A reductive group with maximal torus $T$ has $X^{\ast }=X^{\ast }(T)$ and $X_{\ast }=X_{\ast }(T)$.

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.