## Lie algebras

Last update: 9 September 2012

## Lie algebras

A Lie algebra over a field $F$ is a vector space $𝔤$ over $F$ with a bracket $\left[,\right]:𝔤×𝔤\to 𝔤$ which is bilinear and satisfies

1. $\left[x,y\right]=-\left[y,x\right],$ for all $x,y\in 𝔤$,
2. (Jacobi identity) $\left[x,\left[y,z\right]\right]+\left[z,\left[x,y\right]\right]+\left[y,\left[z,x\right]\right]=0,$ for all $x,y,z\in 𝔤.$
The derived series of $𝔤$ is the sequence

The lower central series of $𝔤$ is the sequence

Let $𝔤$ be a Lie algebra.

1. $𝔤$ is abelian if $\left[𝔤,𝔤\right]=0.$
2. $𝔤$ is nilpotent if ${C}^{n}\left(𝔤\right)=0$ for all sufficiently large $n.$
3. $𝔤$ is solvable if ${D}^{n}\left(𝔤\right)=0$ for all sufficiently large $n.$
4. The radical $\mathrm{rad}\left(𝔤\right)$ is the largest solvable ideal of $𝔤.$
5. The nilradical $\mathrm{nil}\left(𝔤\right)$ is the largest nilpotent?????? ideal of $𝔤.$
6. $𝔤$ is semisimple if $\mathrm{rad}\left(𝔤\right)=0.$
7. $𝔤$ is reductive if $\mathrm{nil}\left(𝔤\right)=0.$
8. A Cartan subalgebra is a maximal abelian subalgebra of semisimple elements.

Then $0⊆nil(𝔤)⊆ rad(𝔤)⊆𝔤$ where $\mathrm{nil}\left(𝔤\right)$ is nilpotent, $\mathrm{rad}\left(𝔤\right)$ is solvable, $𝔤/\mathrm{rad}\left(𝔤\right)$ is semisimple, $\mathrm{rad}\left(𝔤\right)/\mathrm{nil}\left(𝔤\right)$ is abelian, $\mathrm{nil}\left(𝔤\right)$ is nilpotent.

Example [Bou, Chap I, Section 4, Prop 5] The following are equivalent:

1. $𝔤$ is reductive.
2. The adjoint representation of $𝔤$ is semisimple.
3. $\left[𝔤,𝔤\right]$ is a semisimple Lie algebra,
4. $𝔤$ is the direct sum of a semisimple Lie algebra and a commutative Lie algebra.
5. $𝔤$ has a finite dimensional representation such that the associated bilinear form is nondegenerate.
6. $𝔤$ has a faithful finite dimensional representation.
7. $\mathrm{rad}\left(𝔤\right)$ is the center of $𝔤$.

HW: Show that $𝔤$ is reductive if all its representations are completely decomposable.

HW: Show that $𝔤$ is reductive if $𝔤=Z\left(𝔤\right)\oplus \left[𝔤,𝔤\right]$ with $\left[𝔤,𝔤\right]$ semisimple.

The finite dimensional simple Lie algebras over $ℂ$ are

1. (Type ${A}_{n-1}$) $𝔰{𝔩}_{n}\left(ℂ\right)$, for $n\ge 2$,
2. (Type ${B}_{n}$) $𝔰{𝔬}_{2n+1}\left(ℂ\right)$, for $n\ge 1$,
3. (Type ${C}_{n}$) $𝔰{𝔭}_{2n}\left(ℂ\right)$, for $n\ge 1$,
4. (Type ${D}_{n}$) $𝔰{𝔬}_{2n}\left(ℂ\right)$, for $,n\ge 4$, and
5. the five simple Lie algebras ${E}_{6},{E}_{7},{E}_{8},{F}_{4},{G}_{2}$.

The finite dimensional simple Lie algebras over $ℝ$ are ???????????????????????