Coroot

Summary

The coroot of a root $\alpha$ is the vector $\alpha^\vee = 2\alpha/(\alpha, \alpha)$, which rescales $\alpha$ so that the reflection in the hyperplane orthogonal to $\alpha$ reads $x \mapsto x - (x, \alpha^\vee)\alpha$. The coroots of a root system form again a root system, the dual root system. In a semisimple Lie algebra, the coroot of $\alpha$ is an element $h_\alpha$ of the Cartan subalgebra. It lies in the copy of $\mathfrak{sl}_2$ attached to $\alpha$, and $\alpha(h_\alpha) = 2$.