---- > [!definition] Definition. ([[simple Lie algebra]]) > A [[Lie algebra]] $\mathfrak{g}$ is called **simple** if it has no nonzero proper [[ideal of a Lie algebra|ideals]] *and is [[abelian Lie algebra|nonabelian]]* (i.e., also has nonzero bracket). ^definition > [!equivalence] > $\mathfrak{g}$ is simple if and only if $\mathfrak{g}$ is [[abelian Lie algebra|nonabelian]] and the [[adjoint representation]] $\rho$ of $\mathfrak{g}$ is [[irreducible Lie algebra representation|irreducible]]. ^equivalence > [!basicexample] > $\mathfrak{sl}_{2}(\mathbb{C})$ is simple [[classification of the irreps of sl2 over C|because]] its adjoint representation is just the irrep $V(2)$. ^basic-example > [!basicnonexample] Warning. > The nonabelian condition departs from analogy with the definition of [[simple group]]! But we will see why it is good to include here. ^nonexample ---- #### ---- #### 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 > ```