By Wehrfritz B.A.F.
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Extra info for 2-Generator conditions in linear groups
Example text
For the particular Lie algebrsswhich we shall have to consider this also follows from the explicit description to be given below. 1) we obtain a map E : HOmLie(Li,L 2) § HOmBialg(E(Ll),E(L2) ). In fact given a homomorphism ~ : L I + L 2 of Lie algebras there is one and only one homomorphism E(~) : E(L I) § E(L 2) of associative algebras so that L1 - ) L2 F,(~) ECh) co~zm~es, [ ~ECL2) -46- In view of the obvious functorial properties of the maps D, a and ~ associated with each L, with these. B. E(m) will in fact commute In other words E is a functor frgm Lie algebras t o bialgebras.
43 - PROOF (i) => (ii): By Prop. 4 (ii) holds for u = Ai, hence by W R-linearity of ~ for all u E T(Rn). (ii) => (iii) :
In each case it then suffices to verify that the images of the generators di of the algebra E(LF) coincide, and this follows from the explicit description given earlier on. PROOF o_~fTheorem i (D and w Prop. 4). (i) is just Lamina 2. (iii) follows from the fact that F ~-~ LF is a functor, end from (ii). For (ii), we recall (cf. II w Theorem i) that Hom~(F,G) ~ HOmBialg(Un,PF ; Um,PG) Cn = dim F, m = dim G). Recalling the way ~ and Lf were defined, we see now that it suffices to prove that the map 9 HOmBialgCUn,PF ; Um,PG) § HOmLi e (~,L G) is bijective.