---
date: 2022-06-01 20:49
modification date: Wednesday 1st June 2022 20:49:29
title: "fundamental weights"
aliases: [fundamental weight]
---

Last modified date: <%+ tp.file.last_modified_date() %>

---

- Tags: 
	- #lie-theory 
- Refs:
	- #todo/add-references
- Links:
	- [dominant weight](Unsorted/dominant%20weight.md)

---

# fundamental weights

The **fundamental weight**s $\omega_{1}, \ldots, \omega_{n}$ are defined by the property that they form a basis of $\mathfrak{h}_{0}$ dual to the set of [[coroots]] associated to the simple roots. 
That is, the fundamental weights are defined by the condition
$$
2 \frac{\left\langle\omega_{i}, \alpha_{j}\right\rangle}{\left\langle\alpha_{j}, \alpha_{j}\right\rangle}=\delta_{i, j}
$$
where $\alpha_{1}, \ldots \alpha_{n}$ are the [[simple roots]]. 

An element $\lambda$ is then algebraically integral if and only if it is an integral combination of the fundamental weights.

The set of all $\mathfrak{g}$-integral weights is a lattice in $\mathfrak{h}_{0}$ called the [[weight lattice]] for $\mathfrak{g}$, denoted by $P(\mathfrak{g})$.