Fubini-Griffiths-Harris rigidity and lie algebra cohomology

We prove a rigidity theorem for represented semi-simple Lie groups. The theorem is used to show that the adjoint variety of a complex simple Lie algebra g (the unique minimal G orbit in ℙg) is extrinsically rigid to third order (with the exception of g = a1). In contrast, we show that the adjoint variety of SL3ℂ and the Segre product Seg(ℙ1 × ℙn) are flexible at order two. In the SL3ℂ example we discuss the relationship between the extrinsic projective geometry and the intrinsic path geometry. We extend machinery developed by Hwang and Yamaguchi, Se-ashi, Tanaka and others to reduce the proof of the general theorem to a Lie algebra cohomology calculation. The proofs of the flexibility statements use exterior differential systems techniques. © 2012 International Press.

Duke Scholars

Author Colleen M Robles Mathematics