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Is there a classification of embeddings of SL_2 into SP_6 as algebraic groups over Q and R respectively?

see also the link:mathoverflow.net/questions/36762,

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It would help to add a link to your earlier question mathoverflow.net/questions/36762, since the new one asks for a further refinement. To adapt to the groups over the rationals or reals, you probably need to exploit the existence of integral forms for the groups (and their Lie algebras) along with the exponential map. –  Jim Humphreys Aug 31 '10 at 15:24

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You probably want the Jacobson–Morozov theorem, which says that homomorphisms of the Lie algebra sl2 over a field of characteristic 0 to a semisimple Lie algebra g can be classified in terms of the nilpotent elements of g. More precisely, if e, f, h, is the usual basis of sl2 then you can choose the image of e to be any nilpotent element of g, and the images of f and h are then determined up to conjugation by the centralizer of e.

For details see Jacobson's book on Lie algebras.

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in the case of Q ,the homomorphisms between groups is characterized by homomorphisms between lie algebra ? –  TOM Aug 31 '10 at 12:49
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Essentially, at least for simply connected groups. Borel's book on linear algebraic groups should give the exact conditions for this. –  Richard Borcherds Aug 31 '10 at 13:18
    
Thanks you great help –  TOM Aug 31 '10 at 13:49
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There has been helpful discussion of lifting of Lie algebra homomorphisms to simply connected or other groups in an earlier post: mathoverflow.net/questions/36481 –  Jim Humphreys Aug 31 '10 at 15:29
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Assume the base field has characteristic zero. The category of representations of a semisimple Lie algebra is Tannakian, and the algebraic group attached to this category is the simply connected algebraic group with the given Lie algebra. A homomorphism of Lie algebras defines a tensor functor of the Tannakian categories, and hence a homomorphism of the corresponding simply connected algebraic groups. –  JS Milne Sep 1 '10 at 13:08

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