---- > [!definition] Definition. ([[semisimple Lie algebra]]) > A [[Lie algebra]] $\mathfrak{g}$ is called **semisimple** if it factors into [[simple Lie algebra|simple]] [[ideal of a Lie algebra|ideals]]: $\mathfrak{g}=I_{1} \oplus \dots \oplus I_{k}.$ ^definition > [!equivalence] > - $\mathfrak{g}$ is semisimple iff its [[adjoint representation]] is [[faithful Lie algebra representation|faithful]] and [[completely reducible]]. > - [[The Cartan-Killing Criterion|The Cartan-Killing Criterion for semisimplicity]] ^equivalence > [!proof] > $\to$. Suppose $\mathfrak{g}$ semisimple; write $\mathfrak{g}=I_{1} \oplus \dots \oplus I_{k}$ for $I_{i}$ simple ideals. By definition, a simple ideal is an [[irreducible Lie algebra representation|irreducible]] [[Lie algebra subrepresentation|subrepresentation]] of $\text{ad}:\mathfrak{g} \to \mathfrak{gl}(\mathfrak{g})$, thus this factorization witnesses that $\mathfrak{g}$ is completely reducible. As for faithfulness: $\operatorname{ker }\text{ad}=Z(\mathfrak{g})=Z(I_{1}) \oplus \dots \oplus Z(I_{k})$ > where each $Z(I_{i})$ is trivial because nonabelian. (or something am going very fast) > > > > > $\leftarrow$. Using complete reducibility, have $\mathfrak{g}=\mathfrak{g}_{1} \oplus \dots \oplus \mathfrak{g}_{k}$ > for $\mathfrak{g}_{i}$ irreps of $\text{ad}$. As irreducible subrepresentations of $\text{ad}$, the $\mathfrak{g}_{i}$ are ideals of $\mathfrak{g}$ with no nonzero proper ideals. Just need to show they're nonabelian, which follows from faithfulness of $\text{ad}$. > ---- #### ---- #### 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 > ```