----- Analogue for finite [[group|group theory]]: [[Maschke's Theorem]], or [[group-invariant subspace admits group-invariant complement over C]], or something... In fact, this is not really any Lie algebra-specific stuff in this proof. > [!proposition] Proposition. ([[Lie algebra representation completely reducible iff every subrepresentation admits a complement]]) > Let $\mathfrak{g}$ be a [[Lie algebra]]. A [[Lie algebra representation|representation]] $V$ of $\mathfrak{g}$ is [[completely reducible]] if and only if every [[Lie algebra subrepresentation|subrepresentation]] admits [[direct sum of Lie algebras|a]] [[complement of a linear subspace|complementary]] [[Lie algebra subrepresentation|subrepresentation]] $W' \subset V$: $V=W \oplus W'$. ^proposition > [!proof]- Proof. ([[Lie algebra representation completely reducible iff every subrepresentation admits a complement]]) > $\to.$ Assume subrepresentations have complements in $V$. Then to see $V$ is [[completely reducible]], just recursively decompose $V$ into direct sums of $\mathfrak{g}$-invariant subspaces. (This amounts to inducting on $\text{dim }V$.) > > $\leftarrow.$ We didn't prove this direction. ----- #### ---- #### 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 > ```