= Character lattice of a torus
{title2=$X^*(T)=\operatorname{Hom}_{\mathrm{cts}}(T,U(1))$}
For a <compact> <torus> $T\cong U(1)^r$, its <continuous> one-dimensional <characters> form a <free abelian group> of rank $r$. They are precisely $z\mapsto z_1^{m_1}\cdots z_r^{m_r}$ with integer exponents. This follows from the classification of <representations of the circle group>, applied to each factor. These <characters> are the integral <group> <weights> occurring upon restricting a <group representation> to a <maximal torus>; global integrality must be distinguished from integrality against coroots for a <simply connected> semisimple covering <group>.
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