Geodesic
Homomorphisms of Lie Algebras of Algebraic Groups and Analytic Groups
Canadian mathematical bulletin, Tome 38 (1995) no. 3, pp. 352-359

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Let be a Lie algebra homomorphism from the Lie algebra of G to the Lie algebra of H in the following cases: (i) G and H are irreducible algebraic groups over an algebraically closed field of characteristic 0, or (ii) G and H are linear complex analytic groups. In this paper, we present some equivalent conditions for φ to be a differential in the above two cases. That is, φ is the differential of a morphism of algebraic groups or analytic groups as appropriate.In the algebraic case, for example, it is shown that φ is a differential if and only if φ preserves nilpotency, semisimplicity, and integrality of elements. In the analytic case, φ is a differential if and only if φ maps every integral semisimple element of into an integral semisimple element of , where G 0 and H 0 are the universal algebraic subgroups of G and H. Via rational elements, we also present some equivalent conditions for φ to be a differential up to coverings of G in the algebraic case, and for φ to be a differential up to finite coverings of G in the analytic case.
DOI : 10.4153/CMB-1995-051-7
Mots-clés : 17B45, 22E60 
Nahlus, Nazih. Homomorphisms of Lie Algebras of Algebraic Groups and Analytic Groups. Canadian mathematical bulletin, Tome 38 (1995) no. 3, pp. 352-359. doi: 10.4153/CMB-1995-051-7
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