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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.