Timeline for A question on the root systems of simple Lie algebras in the 90 degree case

4 events

when toggle format	what		by	license	comment
Apr 7, 2011 at 17:20	comment	added	Jim Humphreys		Yes, if you refer just to adjacent simple roots; but of course a simple Lie algebra such as $\mathfrak{sl}_4$ can have two orthogonal simple roots corresponding to end nodes of the Dynkin diagram, each connected to a third simple root. What's important is that the relative lengths of roots are well determined, though of course the unit of length you choose can be arbitrary. (In this particular example all roots must have equal length, but ratios of 2 and 3 for squared lengths occur elsewhere in the classification.)
Apr 7, 2011 at 15:14	vote	accept	K McKenzie
Apr 7, 2011 at 15:11	comment	added	K McKenzie		Thank you very much for this! (And yes, I was dimly aware that the angle between the simple roots of a simple Lie algebra is always obtuse, but I had in my mind the whole root diagram here - sorry if that was confusing.) May I ask if it follows from what you're saying that the case in which the roots (i.e. any adjacent roots in the root diagram) are at 90 degrees to one another never corresponds to the root diagram of a simple Lie algebra, whereas those at angles 30, 45 and 60 degrees always do? That would be really useful for me to know as it happens... Thanks again!
Apr 7, 2011 at 13:15	history	answered	Jim Humphreys	CC BY-SA 2.5