Lie algebras

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 1 April 2010

Lie algebras

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

1. $\left[x,y\right]=-\left[y,x\right],$ for all $x,y\in 𝔤,$
2. (Jacobi identity) $\left[x,\left[y,z\right]\right]+z\left[x,y\right]+y\left[z,x\right]=0,$ for all $x,y,z\in 𝔤.$

The derived series of $𝔤$ is the sequence $$D^0𝔤\supseteq D^1𝔤\supseteq \dots ,\text{where}{D}^0𝔤=𝔤\text{ and }{D}^{i+1}𝔤=\left[D^i𝔤,D^i𝔤\right].$$

The lower central series of $𝔤$ is the sequence $$C^1𝔤\supseteq C^2𝔤\supseteq \dots ,\text{where}{C}^0𝔤=𝔤\text{ and }{C}^{i+1}𝔤=\left[𝔤,C^i𝔤\right].$$

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 $\mathrm{nil}\left(𝔤\right)=0.$ 𝔤$is\; reductive\; if\; all\; its\; representations\; are\; completely\; decomposable.$ 𝔤$is\; reductive\; if$ 𝔤=Z\left(𝔤\right)\oplus \left[𝔤,𝔤\right]$with$ \left[𝔤,𝔤\right]$semisimple.$
8. A Cartan subalgebra is a maximal abelian subalgebra of semisimple elements.

Then $$0\subseteq \mathrm{nil}\left(𝔤\right)\subseteq \mathrm{rad}\left(𝔤\right)\subseteq 𝔤$$ 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, and $\mathrm{nil}\left(𝔤\right)$ 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 semisimple Lie algebra,
4. $𝔤$ is the direct sum of a semisimple Lie algebra and a commutative Lie algebra.
7. $𝔤$ 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 $𝔤.$

The finite dimensional simple Lie algebras over $ℂ$ are

1. (Type $A_{n-1}$) $𝔰{𝔩}_n\left(ℂ\right),n\ge 2,$
2. (Type $B_n$) $𝔰{𝔬}_{2n+1}\left(ℂ\right),n\ge 1,$
3. (Type $C_n$) $𝔰{𝔭}_{2n}\left(ℂ\right),n\ge 1,$
4. (Type $D_n$) $𝔰{𝔬}_{2n}\left(ℂ\right),n\ge 4,$ 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 ???????????????????????

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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