Homomorphisms of Lie Algebras of Algebraic Groups and Analytic Groups

Nahlus, Nazih

Canadian mathematical bulletin, Tome 38 (1995) no. 3, pp. 352-359

Résumé

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.

Détail

DOI : 10.4153/CMB-1995-051-7

Mots-clés : 17B45, 22E60