Let $T$ be a compact torus with [Lie algebra](/page/Lie%20Algebra) $\mathfrak t$, and let $\exp_T:\mathfrak t\to T$ denote its Lie exponential map. Define the integral lattice of $T$ by
Let $\ell:\mathfrak t\to\mathbb R$ be a real linear functional, and let $U(1):=\{z\in\mathbb C:|z|=1\}$. Then there exists a character, meaning a continuous [group homomorphism](/page/Group%20Homomorphism) $\lambda:T\to U(1)$, satisfying