----- > [!proposition] Proposition. ([[on factorizing a Lie algebra representation into weight spaces]]) > If $\mathfrak{h}$ is an [[abelian Lie algebra|abelian]] [[Lie algebra]] over $\mathbb{C}$ and $\rho:\mathfrak{h} \to \mathfrak{gl}(V)$ an $\mathfrak{h}$-[[Lie algebra representation|representation]] such that $\rho(x)$ is a [[diagonalizable|semisimple]] [[linear operator]] for all $x \in \mathfrak{h}$, then there exist $\lambda_{1},\dots,\lambda_{k} \in \mathfrak{h}^{*}=\text{Hom}(\mathfrak{h}, \mathbb{C})$ such that $V=\bigoplus_{i=1}^{k}V_{\lambda_{i}}$where $V_{\lambda_{i}}$ denotes the [[simultaneously diagonalizable|simultaneous eigenspace]] $V_{\lambda_{i}}=\{ v \in V: x \cdot v = \lambda_{i}(x) v \text{ for all } x \in \mathfrak{h} \}.$ ^proposition > [!specialization] > The ubiquitous [[root space decomposition of a Lie algebra|root space decomposition]] of a [[Lie algebra]] $\mathfrak{g}$ arises by taking $\rho$ to be the restriction of the [[adjoint representation]] $\text{ad}$ to the [[Cartan subalgebra|CSA]] $\mathfrak{h}:=\mathfrak{t}$, giving $\rho=\text{ad}|_{\mathfrak{t}}:\mathfrak{t} \to \mathfrak{gl}(V=\mathfrak{g})$ . > > More generally, given any [[Lie algebra representation|representation]] $\rho:\mathfrak{g} \to \mathfrak{gl}(V)$ one can restrict $\rho$ to $\mathfrak{h}:=\mathfrak{t} \subset \mathfrak{g}$ and study the resulting decomposition of $V$ into weight spaces. This is discussed e.g. in [[on the weights of a representation]]. ^specialization > [!proof]+ Proof. ([[on factorizing a Lie algebra representation into weight spaces]]) > This is essentially a rehashing of the discussions in [[diagonalizable matrices are simultaneously diagonalizable iff they mutually commute]] and [[simultaneously diagonalizable]]. The abelian and semisimple linear operator conditions are placed to satisfy the hypotheses for the former note, ensuring [[simultaneously diagonalizable|simultaneous diagonalizability]] of all the $\rho$. The latter note uses this to factorize $V$ as claimed. ----- #### ---- #### 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 > ```