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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José Carlos Santos
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Ma Joad
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2roots 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 https://www.youtube.com/watch?v=x95hJ6F87fw&index=11&list=PL6079A8A50EFA181B of Gross's lecture – peter a g Jan 26 '18 at 14:29
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For the corresponding question regarding weights, cf. https://mathoverflow.net/q/154933/27465 – Torsten Schoeneberg Feb 02 '18 at 06:37
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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 $X\mapsto[H,X]$.
José Carlos Santos
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@Stephen You are right, of course. I forgot that convention. – José Carlos Santos Jan 26 '18 at 14:53
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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? – Mike Pierce Feb 28 '18 at 17:34
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