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weight lattice

For a root $\alpha\in\Phi$, define the coroot as $\alpha {}^{ \vee }= {2\alpha \over {\left\langle {\alpha},~{\alpha} \right\rangle}}$.

The root lattice is defined as $$\begin{align*} \Lambda_r \coloneqq{\mathbf{Z}}\Phi \subseteq E \end{align*}$$ for $E$ the Euclidean space in which $\Phi$ lives.

The root lattice is stable under the action of $W$, the Weyl group.

Can be defined as $$\begin{align*} \Lambda_r = \left\{{v\in E {~\mathrel{\Big\vert}~}{(v, e) \over (e,e)} \in {\mathbf{Z}}\quad \forall e\in \Phi }\right\} \end{align*}$$

Dual lattice

The dual lattice in $E$ to the root lattice defined by $$\begin{align*} \Lambda:=\left\{\lambda \in E \mathrel{\Big|}\left\langle\lambda, \alpha^{\vee}\right\rangle \in \mathbb{Z} \text { for all } \alpha \in \Phi\right\} . \end{align*}$$ Here it is enough to let $\alpha$ run over $\Delta$. We call $\Lambda$ the integral weight lattice associated to $\Phi$. It lies in the $\mathbb{Q}$-span $E_{0}$ of the roots in $\mathfrak{h} {}^{ \vee }$and includes the root lattice $\Lambda_{r}$ as a subgroup of finite index.

The root lattice is a subset of the weight lattice.