Advertisement

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
Article

Abstract

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].

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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

Personalised recommendations