---- Let $V$ be a finite-dimensional [[vector space]] over a [[field|field]] $\mathbb{F}$. Let $\mathfrak{gl}(V)$ denote the [[general linear Lie algebra]]. > [!definition] Definition. ([[Lie algebra representation]]) > A **representation** of a [[Lie algebra]] $\mathfrak{g}$ is a [[Lie algebra homomorphism]] $\rho: \mathfrak{g} \to \mathfrak{gl}(V).$ A **$\mathfrak{g}$-action** or **$\mathfrak{g}$-module** is a [[vector space]] $V$ together with a [[bilinear map|bilinear]] pairing $\begin{align} \mathfrak{g} \times V & \to V \\ (x, v) & \mapsto x \cdot v \end{align}$ satisfying $[x,y] \cdot v = x \cdot (y \cdot v) - y \cdot (x \cdot v) \text{ for all } x,y \in \mathfrak{g}, v \in V.$ These two notions have equivalent data: any representation gives rise to a $\mathfrak{g}$-action via $(x,v) \mapsto \rho(x) ( v)$, and every $\mathfrak{g}$-action gives rise to a representation by defining, for all $x \in \mathfrak{g}$, $\rho(x) \in \mathfrak{gl}(V)$ to be the function $v \mapsto x \cdot v$. > $\text{dim }V$ is called the **dimension** or **degree** of the representation. ^definition > [!basicexample] (Representations of $\mathfrak{sl}_{2}(\mathbb{C})$) > > > Let $\mathbb{F}=\mathbb{C}$, $\mathfrak{g}=\mathfrak{sl}_{2}(\mathbb{F})$. [[special linear Lie subalgebra|Recall]] that $\mathfrak{g}$ has the [[basis]] $e,f,g$. > > We have three representations of $\mathfrak{g}$: > > 1. The [[trivial Lie algebra representation]] $\mathfrak{g} \to \mathfrak{gl}_{1}$ > 2. The [[defining representation of a Lie algebra|defining representation]] $\mathfrak{g} \hookrightarrow \mathfrak{gl}_{2}$ > 3. The [[adjoint representation]] $\text{ad: }\mathfrak{g} \to \mathfrak{gl}(\mathfrak{g}) \cong \mathfrak{gl}_{3}$. To determine the matrices, evaluate e.g. $\text{ad}_{e}(e)=0$, $\text{ad}_{e}(h)=[e,h]=-2e$, $\text{ad}_{e}(f)=[e,f]=h$, to get $\text{ad}_{e}= \begin{bmatrix} > 0 & -2 & 0 \\ > 0 & 0 & 1 \\ > 0 & 0 & 0 > \end{bmatrix}$ > where the first column/row is $e$, the second is $h$, the third is $f$; then do the same for the other matrices to get $\text{ad}_{h}=\begin{bmatrix} > 2 & 0 & 0 \\ > 0 & 0 & 0 \\ > 0 & 0 & -2 > \end{bmatrix}$ > ([[diagonal]]!) > and $\text{ad}_{f}= \begin{bmatrix} > 0 & 0 & 0 \\ > -1 & 0 & 0 \\ > 0 & 2 & 0 > \end{bmatrix}$ > ([[lower-triangular matrix|strictly lower-triangular]]!). ^basic-example > [!justification] > Let's quickly justify the equivalence. > > **Representation to $\mathfrak{g}$-action.** [[linear map|Linearity]] in the first argument goes as $(a_{1}x_{1} + a_{2}x_{2}, v) \mapsto \rho(a_{1}x_{1} + a_{2}x_{2} ) v=(a_{1}\rho(x_{1}) + a_{2} \rho(x_{2}))v = a_{1} \rho(x_{1})v + a_{2} \rho(x_{2})v$ > where we have used that $\rho$ is a [[Lie algebra homomorphism]] (in particular, is linear). > > Linearity in second argument goes as $(x, a_{1}v_{1}+a_{2}v_{2})=\rho(x)(a_{1}v_{1}+a_{2}v_{2})$ > and then use that $\rho(x)$ is linear as a map $V \to V$. > > Commutation is checked as $[x,y] \cdot v= \rho([x,y])( v)=[\rho(x), \rho(y)](v)=\rho(x)\rho(y)v-\rho(y)\rho(x)v,$ > where we have used $\rho$ is a [[Lie algebra homomorphism]] (in particular, preserves the Lie bracket). > > **$\mathfrak{g}$-action to representation.** With $\rho$ as defined, we check first that it is [[linear map|linear]]: $\rho(a_{1}x_{1} + a_{2}x_{2})=(a_{1}x_{1}+a_{2}x_{2})\cdot v = a_{1}x_{1} \cdot v + a_{2}x_{2}\cdot v= a_{1} \rho(x_{1}) + a_{1} \rho (x_{2})$ > (that's a little sloppy but basically correct) > (here we used first-argument linearity of the action). We check next that it preserves the bracket: $\begin{align} > \rho([x,y])(v) & = [x,y] \cdot v \\ > & = x \cdot (y \cdot v) - y \cdot ( x \cdot v) \\ > & = \rho(x) \rho(y)(v) - \rho(y) \rho(x)(v) \\ > &= [\rho(x), \rho(y)](v) > \end{align}$ > as required. ---- #### ---- #### 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 > ```