Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Transformation groups resembling the adjoint representation

Author: R. W. Sullivan
Journal: Proc. Amer. Math. Soc. 47 (1975), 491-494
MSC: Primary 57E15
MathSciNet review: 0368054
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: If $ G$ is a compact, connected Lie group, the isotropy subgroups of the adjoint representation of $ G$ are connected and the dimension of the fixed point set of a maximal torus of $ G$ is equal to the the rank of $ G$. Results similar to these are given when $ G$ acts differentiably on an integral cohomology sphere and has the adjoint representation as weak linear model. This is done by analyzing an induced action of the Weyl group of $ G$.

References [Enhancements On Off] (What's this?)

  • [1] Armand Borel, Seminar on transformation groups, With contributions by G. Bredon, E. E. Floyd, D. Montgomery, R. Palais. Annals of Mathematics Studies, No. 46, Princeton University Press, Princeton, N.J., 1960. MR 0116341
  • [2] N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968 (French). MR 0240238
  • [3] Glen E. Bredon, Introduction to compact transformation groups, Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 46. MR 0413144
  • [4] Wu-chung Hsiang and Wu-yi Hsiang, A fixed theorem for finite diffeomorphism groups generated by reflections, The Steenrod Algebra and its Applications (Proc. Conf. to Celebrate N. E. Steenrod’s Sixtieth Birthday, Battelle Memorial Inst., Columbus, Ohio, 1970), Lecture Notes in Mathematics, Vol. 168, Springer, Berlin, 1970, pp. 90–106. MR 0273642
  • [5] Wu-yi Hsiang, On the splitting principle and the geometric weight system of topological transformation groups. I, Proceedings of the Second Conference on Compact Transformation Groups (Univ. Massachusetts, Amherst, Mass., 1971) Springer, Berlin, 1972, pp. 334–402. Lecture Notes in Math., Vol. 298. MR 0380845
  • [6] N. E. Steenrod, Cohomology operations, Lectures by N. E. STeenrod written and revised by D. B. A. Epstein. Annals of Mathematics Studies, No. 50, Princeton University Press, Princeton, N.J., 1962. MR 0145525
  • [7] R. W. Sullivan, Differentiable actions of classical groups on spheres, Topology 9 (1970), 155–167. MR 0261631,
  • [8] R. W. Sullivan, Linear models for compact groups acting on spheres, Topology 13 (1974), 77–87. MR 0339245,

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 57E15

Retrieve articles in all journals with MSC: 57E15

Additional Information

Keywords: Weyl group, adjoint representation, cohomology sphere
Article copyright: © Copyright 1975 American Mathematical Society