By Jörg Wildeshaus

Classically, greater logarithms seem as multivalued services at the projective line. this day they are often interpreted as entries of the interval matrix of a definite version of Hodge constitution, itself known as the "polylogarithm". the purpose of the ebook is to record the sheaf-theoretical foundations of the sphere of polylogarithms. previous, in part unpublished effects and buildings of Beilinson, Deligne, and Levin at the classical and elliptic polylog are generalized to the context of Shimura types. The reader is predicted to have a legitimate history in algebraic geometry. huge elements of the booklet are expository, and meant as a reference for the operating mathematician. the place a self-contained exposition was once impossible, the writer offers references which will make the cloth available for complicated graduate students.

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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).