= Root datum
{wiki}
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> $X^*,X_*$ with a perfect <bilinear map> $\langle-,-\rangle:X^*\times X_*\to\mathbb Z$, finite sets $\Phi\subset X^*\setminus\{0\}$ and $\Phi^\vee\subset X_*\setminus\{0\}$, and a bijection $\alpha\leftrightarrow\alpha^\vee$ satisfying $\langle\alpha,\alpha^\vee\rangle=2$. The associated reflections
$$
s_\alpha(x)=x-\langle x,\alpha^\vee\rangle\alpha,\qquad s_\alpha^\vee(y)=y-\langle\alpha,y\rangle\alpha^\vee
$$
permute the <roots> and <coroots>, respectively, compatibly with the correspondence: $(s_\alpha\beta)^\vee=s_\alpha^\vee(\beta^\vee)$. They generate the <Weyl group>.
For a split <maximal algebraic torus> $T$ in a <reductive algebraic group>, $X^*=\operatorname{Hom}(T,\mathbb G_m)$ and $X_*=\operatorname{Hom}(\mathbb G_m,T)$, paired by composition. For $SL_2$, using $T=\{\operatorname{diag}(t,t^{-1})\}$ gives $X^*=X_*=\mathbb Z$, with root $2$ and coroot $1$. For $PGL_2$ 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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