---- > [!definition] Definition. ([[linear Lie algebra]]) > A [[Lie algebra]] $\mathfrak{g}$ over a [[field]] $\mathbb{F}$ is called **linear** if it can be [[Lie algebra homomorphism|embedded]] as a [[Lie subalgebra]] of the [[general linear Lie algebra]] $\mathfrak{gl}(V)$ for some $\mathbb{F}$-[[vector space]] $V$. ^definition > [!basicexample] > Any [[Lie algebra]] $\mathfrak{g}$ with trivial [[center of a Lie algebra|center]] is linear. Indeed, the [[adjoint representation]] $\begin{align} \text{ad}:\mathfrak{g} &\to \mathfrak{gl}(\mathfrak{g}) \\ x & \mapsto [x,-] \end{align}$ has trivial [[kernel of a Lie algebra homomorphism|kernel]] and is hence an [[injection]]: for any $x \in \mathfrak{g}$, $\text{ad}(x)=0 \implies =[x,y]=0 \ \fa y \in \mathfrak{g} \implies x=0 \text{ (because center is trivial)}.$ ^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 > ```