# radical

Let $\U0001d524$ be a Lie algebra^{}. Since the sum of any two solvable ideals of $\U0001d524$ is in turn solvable, there is a unique maximal solvable ideal of any Lie algebra. This ideal is called the radical^{} of $\U0001d524$. Note that $\U0001d524/\mathrm{rad}\U0001d524$ has no solvable ideals, and is thus semi-simple. Thus, every Lie algebra is an extension (http://planetmath.org/Extension) of a semi-simple algebra by a solvable one.

Title | radical |
---|---|

Canonical name | Radical |

Date of creation | 2013-03-22 13:51:18 |

Last modified on | 2013-03-22 13:51:18 |

Owner | bwebste (988) |

Last modified by | bwebste (988) |

Numerical id | 5 |

Author | bwebste (988) |

Entry type | Definition |

Classification | msc 17B05 |