Radical of a Lie algebra
Let be a field and let be a finite-dimensional Lie algebra over . A maximal solvable ideal, which is called the radical, exists for the following reason.
Firstly let and be two solvable ideals of . Then is again an ideal of , and it is solvable because it is an extension of by . Therefore we may also define the radical of as the sum of all the solvable ideals of , hence the radical of is unique. Secondly, as is always a solvable ideal of , the radical of always exists.
- A Lie algebra is semisimple if and only if its radical is .
- A Lie algebra is reductive if and only if its radical equals its center.
|This algebra-related article is a stub. You can help Wikipedia by expanding it.|