Communications in Mathematical Physics

, Volume 131, Issue 3, pp 495–515 | Cite as

Differential graded Lie algebras and singularities of level sets of momentum mappings

  • William M. Goldman
  • John J. Millson


The germ of an analytic varietyX at a pointxX is said to bequadratic if it is bi-analytically isomorphic to the germ of a cone defined by a system of homogeneous quadratic equations at the origin. Arms, Marsden and Moncrief show in [2] that under certain conditions the analytic germ of a level set of a momentum mapping is quadratic. We discuss related ideas in a more algebraic context by associating to an affine Hamiltonian action a differential graded Lie algebra, which in the presence of an invariant positive complex structure, is formal in the sence of [5].


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  1. 1.
    Abraham, R., Marsden, J.: Foundations of mechanics, second ed. Reading, MA: Addison Wesley 1978Google Scholar
  2. 2.
    Arms, J., Marsden, J., Moncrief, V.: Symmetry and bifurcation of momentum mappings. Commun. Math. Phys.78, 455–478 (1981)Google Scholar
  3. 3.
    Atiyah, M., Bott, R.: The Yang-Mills equations over a compact Riemann surface. Phil. Trans. R. Soc. LondonA308, 523–615 (1982)Google Scholar
  4. 4.
    Deligne, P.: Letter to J. Millson and W. Goldman, April 24, 1986Google Scholar
  5. 5.
    Deligne, P., Griffiths, P. A., Morgan, J. W., Sullivan, D.: Rational homotopy type of compact Kähler manifolds. Invent. Math.29, 245–274 (1975)CrossRefGoogle Scholar
  6. 6.
    Goldman, W. M., Millson, J. J.: Deformations of flat bundles over Kähler manifolds. In: Geometry and Topology, Manifolds, Varieties and Knots. McCrory, C., Shifrin, T. (eds.), Lecture Notes in Pure and Applied Mathematics, vol.105, pp. 129–145. New York, Basel: Marcel Dekker (1987)Google Scholar
  7. 7.
    Goldman, W. M., Millson, J. J.: The deformation theory of representations of fundamental groups of compact Kähler manifolds. Publ. Math. I.H.E.S.67, 43–96 (1988)Google Scholar
  8. 8.
    Goldman, W. M., Millson, J. J.: The homotopy invariance of the Kuranishi space. Ill. J. Math. (memorial issue dedicated to K. T. Chen)Google Scholar
  9. 9.
    Schlessinger, M., Stasheff, J.: Deformation theory and rational homotopy type. Publ. Math. I.H.E.S. (to appear)Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • William M. Goldman
    • 1
  • John J. Millson
    • 1
  1. 1.Department of MathematicsUniversity of MarylandCollege ParkUSA

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