Why is a root system called a "root" system?

Question

Root systems plays an important role in, among other things, classifying semisimple Lie Algebras. Their name suggest that they have something to do with "roots" of a polynomial. Are they the roots of some polynomial? Where does the name "root system" come from?

Answer 1

It comes from the roots of the characteristic polynomial of an endomorphism. If $\mathfrak g$ is a complex semisimple Lie algebra, $\mathfrak h$ is a Cartan subalgebra and $\alpha\in\mathfrak{h}^*$, then $\alpha$ is a root if, for every $H\in\mathfrak h$, $\alpha(H)$ is an eigenvalue of the endomorphism of $\mathfrak g$ defined by $X\mapsto[H,X]$.