---
date: 2022-06-01 20:50
modification date: Wednesday 1st June 2022 20:50:49
title: "weight lattice"
aliases: [weight lattice, root lattice]
---

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---

- Tags: 
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- Refs:
	- #todo/add-references
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# weight lattice

For a root $\alpha\in\Phi$, define the [[coroot]] as $\alpha\dual = {2\alpha \over \inp{\alpha}{\alpha}}$.

The **root lattice** is defined as 
$$
\Lambda_r \da \ZZ\Phi \subseteq E
$$
for $E$ the Euclidean space in which $\Phi$ lives. 

The root lattice is stable under the action of $W$, the [Weyl group](Unsorted/Weyl%20group.md).

Can be defined as 
$$
\Lambda_r = \ts{v\in E \st {(v, e) \over (e,e)} \in \ZZ \quad \forall e\in \Phi }
$$

## Dual lattice

The dual lattice in $E$ to the root lattice defined by
$$
\Lambda:=\left\{\lambda \in E \mid\left\langle\lambda, \alpha^{\vee}\right\rangle \in \mathbb{Z} \text { for all } \alpha \in \Phi\right\} .
$$
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 $\mathfrak{h}\dual$and includes the root lattice $\Lambda_{r}$ as a subgroup of finite index. 

The root lattice is a subset of the weight lattice.