---- [^1]: Here we are using the fact that the [[tensor product of modules]] is [[submodule generated by a subset|spanned]] by [[tensor product of modules|pure tensors]]. > [!definition] Definition. ([[tensor product of Lie algebra representations]]) > Let $\mathfrak{g}$ be a [[Lie algebra]] and $V,W$ $\mathfrak{g}$-[[Lie algebra representation|representations]]. Their [[tensor product of modules|tensor product]] $V \otimes W=\text{span}\{ v \otimes w \}_{v \in V, w \in W}$ [^1] is a $\mathfrak{g}$-representation via [[linear maps and basis of domain|linear extension]] of $x \cdot (v \otimes w):=(x \cdot v) \otimes w + v \otimes (x \cdot w).$ ('transfer $x$ to the right') > This induces $\mathfrak{g}$-actions on the [[symmetric power]] $\text{Sym}^{n}(V)$ and [[exterior power]] $\Lambda^{n}(V)$. As with [[vector space|vector spaces]], one has $V \otimes V \cong \text{Sym}^{2}(V) \oplus \Lambda^{2}(V)$ for representations. > [!basicexample] > We have $V^{*} \otimes W \cong \text{Hom}(V,W)$ as $\mathfrak{g}$-[[Lie algebra representation|representations]]. Indeed, [[isomorphism for tensor product and homset for finite-dimensional spaces|from this note]] we already know they are [[linear isomorphism|isomorphic]] as [[vector space|vector spaces]]. So we just have to check that the map $\Phi$ defined in that note is in fact a [[morphism of Lie algebra representations]], i.e., is $\mathfrak{g}$-equivariant. ^basic-example ---- #### ---- #### 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 > ```