The Generalised Jacobson-Morosov Theorem

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The author considers homomorphisms H \to K from an affine group scheme H over a field k of characteristic zero to a proreductive group K. Using a general categorical splitting theorem, AndrU and Kahn proved that for every H there exists such a homomorphism which is universal up to conjugacy. The author gives a purely group-theoretic proof of this result. The classical Jacobson-Morosov theorem is the particular case where H is the additive group over k. As well as universal homomorphisms, the author considers more generally homomorphisms H \to K which are minimal, in the sense that H \to K factors through no proper proreductive subgroup of K. For fixed H, it is shown that the minimal H \to K with K reductive are parametrised by a scheme locally of finite type over k.