---- > [!definition] Definition. ([[defining representation of a Lie algebra]]) > Let $\mathfrak{g}$ be a [[Lie algebra]] and let $V$ be a [[vector space]] over a [[field|field]] $\mathbb{F}$. If $\mathfrak{g}$ is defined as a [[Lie subalgebra]] of the [[general linear Lie algebra]] $\mathfrak{gl}(V)$, then the [[inclusion map]] is a [[Lie algebra homomorphism]] $\rho:\mathfrak{g} \to \mathfrak{gl}(V)$, hence a [[Lie algebra representation]] of $\mathfrak{g}$. It is called the **defining representation**. ^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 > ```