---- > [!definition] Definition. ([[direct sum of Lie algebras]]) > Let $\mathfrak{g}$ be a [[Lie algebra]] over a [[field]] $\mathbb{F}$. Let $V,W$ be two $\mathfrak{g}$-[[Lie algebra representation|representations]]. Their [[direct sum of modules|direct sum]] (as [[vector space|vector spaces]]) $V \oplus W$ naturally carries the structure of a $\mathfrak{g}$-[[Lie algebra representation|representation]] by component-wise action: $x \cdot (v,w)=(x\cdot v, x \cdot w).$ ^definition ---- #### ---- #### 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 > ```