---- > [!definition] Definition. ([[root and weight lattice of a root system]]) > Let $(\Phi, E)$ be a [[root system]]. [[lattice (groups)|Recall]] that a subset $L \subset E$ is a lattice if it is the $\mathbb{Z}$-[[submodule generated by a subset]] of an $\mathbb{R}$-basis of $E$. > > The **root lattice** $\mathbb{Z} \Phi$ is by definition the $\mathbb{Z}$-span of $\Phi$ in $E$. The **weight lattice** $X$ [[reflection|is the set]] $X=\{ \lambda \in E : \langle \lambda,\check \alpha \rangle \in \mathbb{Z} \text{ for all } \alpha \in \Phi \}.$ > These are indeed lattices. Fixing a [[root basis]] $\Delta=\{ \alpha_{1},\dots,\alpha_{\ell} \}$ of $\Phi$, we have $\mathbb{Z}\Phi=\mathbb{Z} \alpha_{1} \oplus \dots \oplus \mathbb{Z} \alpha_{\ell}$, so $\mathbb{Z}\Phi$ is. As for $X$, let $\omega_{1},\dots \omega_{\ell}$ be the unique elements of $E$ satisfying $\langle \omega_{i}, \check \alpha_{j} \rangle =\delta_{ij}.$ > These are called the **fundamental weights**.[^3] As an exercise, can check $X=\{ \lambda \in E : \langle \lambda, \check \alpha_{i} \rangle \in \mathbb{Z} \text{ for all }i \}.$ > Therefore, $X=\mathbb{Z} \omega_{1} \oplus \dots \oplus \mathbb{Z} \omega_{\ell}$. > > Since $\langle \alpha, \check \beta \rangle \in \mathbb{Z}$ for all $\alpha, \beta \in \Phi$, $\mathbb{Z}\Phi \subset X$. In general this inclusion is strict. The [[quotient group|quotient]] $X / \mathbb{Z}\Phi$ is a finite[^2] [[group]], called the **fundamental group**[^1] of $\Phi$. $|X / \mathbb{Z} \Phi|$ equals the [[determinant]] of the [[cartan matrix]]. > > [!basicexample] > The fundamental weights for $A_{2}$ are pictured in green below. General weights are $\mathbb{Z}$-linear combinations thereof. > ![[Pasted image 20250514190548.png]] > The fundamental weights for $B_{2}$ are pictured in green below. > ![[Pasted image 20250514192009.png]] > Recall that $C_{2}$ is [[root system isomorphism|isomorphic]] to $B_{2}$, so not much more to say there. As for $G_{2}$, its root and weight lattice are the same (not hard to see; in a sense, $G_{2}$ is what you get if you take $\Phi=A_{2}$ and adjoint the fundamental weights of $A_{2}$ plus their reflections). ^basic-example > [!note] Remark. > The terminology *weight* lattice might initially seem odd. The object defined here can be defined for any [[root system]]. Where are the 'weights'? The terminology comes from the [[Lie algebra representation|representation theory of]] [[Lie algebra|Lie algebras]], where the [[on the weights of a representation|weights of a representation]] $V$ of a [[Lie algebra]] $\mathfrak{g}$ live in the weight lattice of the [[root system of a Lie algebra|root system associated to]] $\mathfrak{g}$. ^note ---- #### [^1]: The [[fundamental group|naming]] is not a coincidence. [^2]: This is a general fact: if $L_{1} \subset L_{2}$ are lattices of $E$ then $L_{2} / L_{1}$ is finite. [^3]: Geometrically (recalling how $\langle -, \check{-} \rangle$ is defined), $\omega_{i}$ is the unique element of $E$ whose projection onto $\alpha_{i}$ equals $\frac{1}{2}\alpha_{i}$ and which lives in the hyperplane $H_{\alpha_{j}}=\text{span}^{\perp}\alpha_{j}$ for all $j \neq i$. ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```