By Ian Stewart (auth.)
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Extra resources for Lie Algebras
To conclude this section, we state the classification in full and explain how much of it we have proved. Theorem 8,~ Over a field k, algebraically aimensional closed, of characteristic simple Lie algebras are in 1-1 correspondence 0, finite- with their 57 stars, which are of the types A ~ ( ~ A l} B ~ C ~ A 2) C~(~ A 3) D~(~A 4) G 2, F 4, E 6 , E 7, E 8 • Finite-dimensional semislmple Lie algebras are direct sums L = L 1 @ ... @ L 2 of simple Lie algebras. Such an expresmion is unique up to the order of the direct factors.
This reduces us to consider, with this choice of x,y,z, three cases. (1) ~ (il) v (iii) ~ +~+y = = 0 -~,~ ~ ± + O+y is a root -6, and ~,O,y,5 does not split up as two pairs of opposite roots. c~,e ] + Ny,~[e_~eB] SO that N ,ohy + N o , y h + Ny, h~ = O. But h +h~+hy = 0 and h,h~ are linearly independent so substituting for hy and collecting coefficients we get (2) (ii) = 0 NO = NO,y Roots are ~,-~,~. = Ny,~ in this case. Let x = e , y = e_~, z = e~. 4~. Put M~,~ = N~,~ -N_~,_~. Then this gives Ma,~ = Ma,~-~ = (h~,h~).
The star-vectors are images of the Pi under elements of W, so they are determined as well. the B-chains, and hence ( h , h ~ ) / ( h , h ) We can then determine = (s-t)/2 is determined for all roots ~,~ . (h,h~) Then ll(h ~ ,h ~ ) =Z (ha,h~)/(h ~ ~ ,h ) is determined, k is determined. so A general Caftan matrix will consist of blocks down the diagonal, each block being one of the above Caftan matrice~, with O's everywhere else. To complete the classification of semisimple Lie algebras we have to show several things: The Cartan matrix determines a unique semisimple Lie algebra (up to isomorphism).