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Hello, I have recently started reading about Lie algebras. However all the examples I have encountered so far are simple and semisimple Lie algebras. Thus I would love to see an example of a real or complex finite dimensional Lie algebra $A$ with the following property :

$A$ is non abelian and it contains non trivial ideals.

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Usually, the difficulty is in finding the simple ones :) –  Mariano Suárez-Alvarez Oct 14 '11 at 17:13
    
Community-wiki? Examples come in all shapes and sizes. –  Jim Humphreys Oct 14 '11 at 20:43
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@ Srifo B: Here is an exercise for you: given simple Lie algebras $s_1,...,s_n$, construct a Lie algebra with abelian radical $r$ and semi-simple part $s_1\times...\times s_n$, such that $[r,s_i]\neq 0$ for $i=1,...,n$. (Hint: think of the adjoint representation). –  Alain Valette Oct 15 '11 at 8:00

5 Answers 5

A nice example to play around with is the Lie algebra of upper triangular matrices. It is solvable, so has plenty of ideals and things like that.

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And if you take the strictly upper triabgular, then it is nilpotent. –  Yiftach Barnea Oct 14 '11 at 19:01

I think it is also a good idea to take a look at the following papers by Willem de Graaf et al. about nilpotent and solvable Lie algebras of small dimension over arbitrary fields:

http://arxiv.org/abs/math/0404071

http://arxiv.org/abs/math/0511668

http://arxiv.org/abs/1011.0361

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The 3-dimensional Heisenberg Lie algebra can be described by the presentation:

$$\mathcal{H}=\big\langle x, y, z\,\big\vert\,[x,y] = z, [x,z]=[y,z]=0\big\rangle$$

The derived subalgebra $[\mathcal{H},\mathcal{H}]$ is a central ideal spanned by $z$, and the whole Lie algebra is a nilpotent Lie algebra (thus not simple or semi-simple).

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Take any non-abelian Lie algebra $L$ and consider $L\oplus {\mathbb C}^n$. This Lie algebra is non-abelian, and non-semisimple because it has a non-trivial radical.

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To get a good idea about the relative paucity of simple Lie algebras (as Mariano says in his comment above), you could take a look at a list of low-dimensional Lie algebras. For instance, here: J. Patera,R.T. Sharp,P. Winternitz, and H. Zassenhaus, Invariants of real low dimension Lie algebras, J. Mathematical Phys. 17 (1976), no. 6, 986–994.

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