radical

Let $\mathfrak{g}$ be a Lie algebra. Since the sum of any two solvable ideals of $\mathfrak{g}$ is in turn solvable, there is a unique maximal solvable ideal of any Lie algebra. This ideal is called the radical of $\mathfrak{g}$ . Note that $\mathfrak{g}/\mathrm{rad}\,\mathfrak{g}$ 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