---- > [!definition] Definition. ([[irreducible Lie algebra representation]]) > Let $\mathfrak{g}$ be a [[Lie algebra]] and $\rho:\mathfrak{g} \to \mathfrak{gl}(V)$ a [[Lie algebra representation|representation]] of $\mathfrak{g}$. $(\rho, V)$ is said to be **irreducible** if $V \neq 0$ and it has no proper nonzero [[Lie algebra subrepresentation|subrepresentations]]. Otherwise we call $(\rho, V)$ **reducible**. ^definition- > [!basicexample] > The three representations of $\mathfrak{sl}_{2}(\mathbb{C})$ [[Lie algebra representation#^basic-example|found here]] are all irreps. ^basic-example Trivial rep immediate. Defining rep: if $W$ is a proper subrep, $\text{dim }W=1$, i.e., $W=\text{span }w$ for some $w \in \mathfrak{gl}_{2}$. Consider the matrix multiplication $e \cdot w=e w=\begin{bmatrix} w_{11} & w_{12} \\ 0 & 0 \end{bmatrix},$ definitely not a nonzero scalar multiple of $w$. Adjoint rep: if $W$ is a proper subrep, $\text{dim }W \in \{ 1,2 \}$. First suppose $\text{dim }W = 1$, i.e., $W=\text{span }w$ for some $w \in \mathfrak{sl}_{2}$, $w=w_{e}e+w_{h}h+w_{f}f$. Observe $e \cdot w=-2 w_{h}e + h w_{f},$ definitely not a nonzero scalar multiple of $w$. Next suppose $\text{dim }W=2$. At this point it's easiest to just use the classification: $\{ e,h,f \}$ [[diagonalizable|diagonalize]] $[h,-]$, yielding [[weights characterize any representation of sl2(C)|weights]] $\{\{ 2,0,-2 \}\}$. By complete reducibility of $\mathfrak{sl}_{2}(\mathbb{C})$, $\text{ad}$ is a direct sum of irreps. One of those irreps, $V(2)$, takes $2,0,-2$ with it. There are no weights left, hence no other summands. This forces $V(2)$ to be the adjoint rep. ---- #### ---- #### 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 > ```