Root datum of a compact connected Lie group

Assume the Axiom of Choice. A reduced compact root datum is a quadruple $(X,\mathrm{\Phi },X^{\vee },{\mathrm{\Phi }}^{\vee })$ in which:

1. $X$ and $X^{\vee }$ are finite free abelian groups of the same rank, equipped with a perfect $\mathbb{Z}$-bilinear pairing $⟨\cdot ,\cdot ⟩:X\times X^{\vee }\to \mathbb{Z}$;
2. $\mathrm{\Phi }\subseteq X$ and ${\mathrm{\Phi }}^{\vee }\subseteq X^{\vee }$ are finite subsets with a fixed bijection $\mathrm{\Phi }\to {\mathrm{\Phi }}^{\vee }$, $\alpha \mapsto {\alpha }^{\vee }$, such that $⟨\alpha ,{\alpha }^{\vee }⟩=2$ for every root;
3. the paired reflections $s_{\alpha }(x)=x-⟨x,{\alpha }^{\vee }⟩\alpha ,s_{\alpha }^{\vee }(y)=y-⟨\alpha ,y⟩{\alpha }^{\vee }$ preserve $\mathrm{\Phi }$ and ${\mathrm{\Phi }}^{\vee }$ and are compatible: if $s_{\alpha }(\beta )=\gamma$ then $s_{\alpha }^{\vee }({\beta }^{\vee })={\gamma }^{\vee }$;
4. $\mathrm{\Phi }$ is reduced: if $\alpha ,c\alpha \in \mathrm{\Phi }$ for $c\in \mathbb{Z}$, then $c=\pm 1$ (equivalently, no root is a nontrivial integral multiple of another).

For a compact connected Lie group $G$ with maximal torus $T$, the root datum of $(G,T)$ is $(X,\mathrm{\Phi },X^{\vee },{\mathrm{\Phi }}^{\vee })=(X^{\ast }(T),\mathrm{\Phi }(G,T),X_{\ast }(T),{\mathrm{\Phi }}^{\vee }),$ where $X^{\ast }(T)$ and $X_{\ast }(T)$ are the character and cocharacter lattices (Character and cocharacter lattices), $\mathrm{\Phi }(G,T)$ is the root system of the pair (Roots of a compact connected Lie group), and ${\mathrm{\Phi }}^{\vee }=\{{\alpha }^{\vee }\}$ consists of the cocharacters supplied by the compact root $SU(2)$ subgroups, so that $⟨\alpha ,{\alpha }^{\vee }⟩=2$ (Analytic and root-system Weyl groups agree). The pairing is the perfect character–cocharacter pairing of the definition above; the reflections are preserved by the identification of the root Weyl group with $W(G,T)$.

Central torus directions are retained: the roots vanish on $Z(G{)}^0$ and do not span $X$ when the centre is positive-dimensional, and the datum records the central character lattice as part of $X$ rather than discarding it.

• Two root data are isomorphic when there are isomorphisms of $X$ and $X^{\vee }$ preserving the pairings and the root and coroot sets; identifying the source of the isomorphism is the exact sense in which the classification by root data holds on this page.
• The reflection formula in (3) is the abstract form of the geometric reflection $s_{\alpha }(\lambda )=\lambda -⟨\lambda ,{\alpha }^{\vee }⟩\alpha$ of the root systems page, written additively for the lattice $X$.
• The definition of a root datum here is deliberately symmetric in $X$ and $X^{\vee }$; the perfect pairing is data, not a consequence of the axioms.
• Empty root and coroot sets are allowed. In particular, a torus has root datum $(X,\mathrm{\varnothing },X^{\vee },\mathrm{\varnothing })$; the root-indexed conditions above are then vacuous.