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?

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    $\begingroup$ roots of a characteristic poly of $t$ in a torus acting on the lie algebra- see almost exactly 5 minutes on, from approx the 15:00 mark in youtube.com/… of Gross's lecture $\endgroup$
    – peter a g
    Jan 26, 2018 at 14:29
  • $\begingroup$ For the corresponding question regarding weights, cf. mathoverflow.net/q/154933/27465 $\endgroup$ Feb 2, 2018 at 6:37

1 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 $X\mapsto[H,X]$.

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    $\begingroup$ ...except that, usually by fiat, $0$ is not a root. $\endgroup$
    – Stephen
    Jan 26, 2018 at 14:50
  • $\begingroup$ @Stephen You are right, of course. I forgot that convention. $\endgroup$ Jan 26, 2018 at 14:53
  • $\begingroup$ Is there a way to describe that same characteristic polynomial without referencing semisimple Lie algebras? Like, just starting with the axioms of a root system? Or starting with a Dynkin/Euclidean diagram and it's corresponding Cartan/Tits form? $\endgroup$ Feb 28, 2018 at 17:34
  • $\begingroup$ I've asked my previous comment as a separate question here. $\endgroup$ Mar 2, 2018 at 20:33

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