Journal of Differential Geometry

Lie group valued moment maps

Anton Alekseev, Anton Malkin, and Eckhard Meinrenken

Full-text: Open access

Article information

J. Differential Geom., Volume 48, Number 3 (1998), 445-495.

First available in Project Euclid: 26 June 2008

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58F05
Secondary: 55N91: Equivariant homology and cohomology [See also 19L47] 57S25: Groups acting on specific manifolds


Alekseev, Anton; Malkin, Anton; Meinrenken, Eckhard. Lie group valued moment maps. J. Differential Geom. 48 (1998), no. 3, 445--495. doi:10.4310/jdg/1214460860.

Export citation


  • [1] A. Yu. Alekseev, On Poisson actions of compact Lie groups on symplectic manifolds, J. Differential Geom. 45 (1997) 241-256.
  • [2] A. Yu. Alekseev and A. Z. Malkin, Symplectic structure of the moduli space of at connections on a Riemann surface, Comm. Math. Phys. 169 (1995) 99-119.
  • [3] M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London 308 (1982) 523-615.
  • [4] S. K. Donaldson, Boundary value problems for Yang-Mills elds, J. Geom. Phys. 8 (1992) 89-122.
  • [5] S. K. Donaldson and P. Kronheimer, The geometry of four-manifolds, Oxford Math. Monographs, 1990.
  • [6] V. G. Drinfeld, Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equation, Soviet. Math. Dokl. 27 (1983) 68-71.
  • [7] H. Flaschka and T. Ratiu, A convexity theorem for Poisson actions of compact Lie groups, Ann. Sci. Ecole Norm. Sup. 29 (1996) 787-80
  • [8] V. V. Fock and A. A. Rosly, Poisson structure on moduli of at connections on Riemann surfaces and r-matrix, Preprint, 1992.
  • [9] W. Goldman, The symplectic nature of fundamental groups of surfaces, Adv. Math. 54 (1984) 200-225.
  • [10] K. Guruprasad, J. Huebschmann, L. Je rey, and A. Weinstein, Group systems, groupoids, and moduli spaces of parabolic bundles, Preprint, 1995.
  • [11] J. Huebschmann, Poisson structure on certain moduli spaces for bundles on a surface, Ann. Inst. Fourier (Grenoble) 45 (1995) 65-91.
  • [12] L. Je rey, Extended moduli spaces of at connections on Riemann surfaces, Math. Ann. 298 (1994) 667-692.
  • [13] L. Je rey, Group cohomology construction of the cohomology of moduli spaces of at connections on 2-manifolds, Duke Math. J. 77 (1995) 407-429.
  • [14] Y. Karshon, An algebraic proof for the symplectic structure of moduli space, Proc. Amer. Math. Soc. 116 (1992) 591-605.
  • [15] C. King and A. Sengupta, A symplectic structure on surfaces with boundary, Comm. Math. Phys. 175 (1996) 657-671.
  • [16] J.-H. Lu, Momentum mappings and reduction of Poisson actions, Symplectic geometry, groupoids, and integrable systems (Berkeley, CA, 1989), Math. Sci. Res. Inst. Publ., Vol. 20, Springer, New York, 1991, 209-226.
  • [17] J.-H. Lu and A. Weinstein, Poisson-Lie groups, dressing transformations and Bruhat decompositions, J. Differential Geom. 31 (1990) 501-526.
  • [18] D. McDu, The moment map for circle actions on symplectic manifolds, J. Geom. Phys. 5 (1988) 149-160.
  • [19] E. Meinrenken and C. Woodward, A symplectic proof of Verlinde factorization, Preprint, August 1996.
  • [20] E. Meinrenken and C. Woodward, Fusion of Hamiltonian loop group manifolds and cobordism, Preprint, July 1997.
  • [21] A. Pressley and G. Segal, Loop groups, Oxford University Press, Oxford, 1988.
  • [22] E. Verlinde, Fusion rules and modular transformations in 2D conformal eld theory, Nuclear Phys. B 300 (1988) 360-376.
  • [23] A. Weinstein, The symplectic structure on moduli space, The Floer memorial volume, Progr. Math., 133, Birkhauser, Basel, 1995, 627-635.
  • [24] J. Weitsman, A Duistermaat-Heckman formula for symplectic circle actions, Internat. Math. Res. Notices 12 (1993) 309-312.
  • [25] E. Witten, On quantum gauge theories in two dimensions, Comm. Math. Phys. 141 (1991) 153-209. Uppsala University, Sweden Yale University University of Toronto