nLab
simple Lie algebra

Definition

A simple Lie algebra is a Lie algebra $𝔤$ such that:

• $𝔤$ is not abelian, and
• the only proper ideal of $𝔤$ is the zero ideal.

Equivalently, a simple Lie algebra is a simple object of LieAlg that also is nonabelian. Note that there are only two abelian Lie algebras whose only proper ideal is the zero ideal: the trivial Lie algebra (which is not a simple object in $\mathrm{Lie}\mathrm{Alg}$ either, since the zero ideal is not proper either) and the line (which is a simple object in $\mathrm{Lie}\mathrm{Alg}$ but is still not considered a simple Lie algebra).

Classification

Simple Lie algebras over an algebraically closed field of characteristic zero, like many other things in mathematics, may be classified by Dynkin diagram?s. We have:

• ${𝔞}_n={\mathrm{𝔰𝔩}}_{n+1}$, the special linear Lie algebra? of rank $n$. We count this only for $n\ge 1$, since ${𝔞}_0$ is the trivial Lie algebra (which is not simple but is still semisimple).

• ${𝔟}_n={\mathrm{𝔰𝔬}}_{2n+1}$, the odd-dimensional special orthogonal Lie algebra of rank $n$. We count this only for $n\ge 2$, since ${𝔟}_n={𝔞}_n$ for $n<2$.

• ${𝔠}_n={\mathrm{𝔰𝔭}}_n$, the symplectic Lie algebra? of rank $n$. We count this only for $n\ge 3$, since ${𝔠}_n={𝔟}_n$ for $n<3$.

• ${𝔡}_n={\mathrm{𝔰𝔬}}_{2n}$, the even-dimensional special orthogonal Lie algebra of rank $n$. We count this only for $n\ge 4$, since ${𝔡}_n={𝔞}_n$ for $n<2$ and $n=3$, while ${𝔡}_2={𝔞}_2\oplus {𝔞}_2$ (which is not simple but is still semisimple).

• ${𝔢}_n$, an exceptional Lie algebra? that only exists for rank $n<9$. We count this only for $n\ge 6$ (thus for $n=6,7,8$ in all), since ${𝔢}_5={𝔡}_5$, ${𝔢}_4={𝔞}_4$, ${𝔢}_n={𝔞}_{n-1}\oplus {𝔞}_1$ (which is not simple but is still semisimple) for $2\le n<4$, and ${𝔢}_n={𝔞}_n$ for $n<2$.

• the exceptional Lie algebra?s ${𝔣}_4$ and ${𝔤}_2$, which exist only for those ranks.

If you want to classify simple objects in $\mathrm{Lie}\mathrm{Alg}$, then there is one other possibility: the line (which has no corresponding Dynkin diagram).

It is much more difficult to classify simple Lie algebras over non-closed fields, over fields with positive characteristic, and especially over non-fields.

Semisimple Lie algebras

A semisimple Lie algebra is a direct sum of simple Lie algebras. In particular, every simple Lie algebra is semisimple, but there are many more.

Simple Lie groups

A Lie group is a simple Lie group if the Lie algebra corresponding to it under Lie integration is simple.