# "Semi-simplicity" of Lie algebra elements.

Why are diagonalizable elements of Lie algebra called "semi-simple"?
Is there a notion of "simple" elements? Is it related to "semi-simplicity" of the Lie algebra?

• I don't know much about lie algebras but there is a general concept of simplicity and semi-simplicity for modules and algebras over modules (a lie algebra is an algebra over a field with additional properties). Check the chapter VI of the book Algèbres et Modules (I. Assem) for further information..
– PtF
Jul 25, 2015 at 12:34
• The first question has been answered here. Jul 25, 2015 at 20:13

Consider an $n\times n$ matrix $A$ over the complex numbers. Then $A$ defines a structure of $\mathbb{C}[X]$-module over $\mathbb{C}^n$, by letting $X$ act by $A$. Now, this module is semi-simple (that is, it is a direct sum of simple (or irreducible) modules) if and only if the matrix $A$ is diagonalizable. Indeed, $A$ is conjugate to a Jordan normal form, and each Jordan block defines an indecomposable direct summand of the module. It is not hard to see that a Jordan block defines a simple module if and only if it is one-dimensional. Hence the module is semi-simple if and only if all Jordan blocks have size 1, which means that $A$ is diagonalizable.