engleski

Distinguished nilpotent orbits, Kostant pairs and normalizers of Lie algebras

A pair of Lie algebras (g, g_1) will be called a Kostant pair if g is semisimple, g_1 is reductive in g and the restriction of the Killing form B_g to g_1 is nondegenerate. We study the class of such (nonsymmetric) pairs and obtain some useful and new structural results. We study the structure of the normalizers N_g(g_1), and as a consequence we obtain some corresponding worthy results about algebraic groups. In particular we consider an interesting case when g_1 is a distinguished sl_2-subalgebra of g. Conmbined with the research due to V. L. Popov we observe that the notions of self-normalizing (reductive) subalgebras of a semisimple Lie algebra and projective self-dual algebraic subvarieties of the usual nilpotent cones are closely related.

semisimple Lie algebra; Cartan subalgebra; root; root system; Borel subalgebra; pair of Lie algebras; Kostant pair; normalizer; self-normalizing subalgebra; nilpotent element; distinguished nilpotent element; nilpotent orbit

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