ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

7. Lie Groups and Lie Algebras

1991 MS Classification: 17, 22 Burde, Dietrich, University of Düsseldorf Affine and projective structures on Lie groups This talk presents a conjecture in the theory of affinely and projectively flat manifolds. In 1977 J. Milnor asked: Does every solvable Lie group G admit a complete left-invariant affine structure ? The question became a famous conjecture. It was disproved recently by Benoist and by Grunewald and the author. There are filiform nilpotent Lie groups without any leftinvariant affine structure. The only known examples are of dimension 10 and 11. To prove this a refinement of Ado’s Theorem for Lie algebras is needed. A positive result holds for classes of filiform Lie groups of dimension n ≥ 12 which corresponds to certain irreducible components of the variety of nilpotent Lie algebras. Reference: D. Burde: Affine structures on nilmanifolds, Int. J. of Math. 7 (1996), 599-616.