---- > [!definition] Definition. ([[ideal of a Lie algebra]]) > Let $\mathfrak{g}$ be a [[Lie algebra]] over [[field]] $\mathbb{F}$. An **ideal** of $\mathfrak{g}$ is a [[linear subspace]] $I$ of $\mathfrak{g}$ satisfying $[\mathfrak{g}, I] \subset I$, that is, for all $x \in \mathfrak{g}, i \in I$, one has $[x,i] \in I$. > This is a stronger condition than being a [[Lie subalgebra]]. In particular, note that every ideal is a [[Lie subalgebra|subalgebra]] — compare to the notion of [[ideal]] in [[ring|ring theory]]. ^definition > [!equivalence] > An ideal is precisely a [[Lie algebra subrepresentation|subrepresentation]] of the [[adjoint representation]] $\text{ad}:\mathfrak{g} \to \mathfrak{gl}(\mathfrak{g})$. ^equivalence > [!intuition] > The analogy between [[Lie algebra|Lie algebras]] and [[Lie group|Lie groups]] has $\begin{align} \text{subalgebra} &\leftrightarrow \text{subgroup} \\ \text{ideal} & \leftrightarrow \text{normal subgroup} \end{align}$ (indeed, the Lie bracket is a bit reminiscent to [[conjugate|conjugation]]). ^intuition - [ ] the usual [[kernel iff ideal]] thing holds, FIT variant, [[universal property]], all that routine ---- #### ---- #### 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 > ```