# 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 Radical 2013-03-22 13:51:18 2013-03-22 13:51:18 bwebste (988) bwebste (988) 5 bwebste (988) Definition msc 17B05