Heckman, Kostant, and Steinberg formulas for symplectic manifolds

Guillemin, Victor; Prato, Elisa

doi:10.1016/0001-8708(90)90087-4

Cited by 31 publications

(40 citation statements)

References 10 publications

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“…Singular reduction is closely related to convexity properties of the momenum map. 42,45 The detailed structure of J Ϫ1 (0)/G for compact Lie groups acting on finite dimensional manifolds was developed in Sjamaar and Lerman ͓1991͔ 146 and extended for proper Lie group actions to J Ϫ1 (O )/G by Bates and Lerman ͓1997͔, 13 if O is locally closed in g*. Ortega ͓1998͔ 129 and Ortega and Ratiu ͓2000͔ 128 redid the entire singular reduction theory for proper Lie group actions starting with the point reduced spaces J Ϫ1 ()/G and also connected it to the more algebraic approach to reduction theory of Arms, Cushman, and Gotay ͓1991͔.…”

mentioning

confidence: 99%

Reduction theory and the Lagrange–Routh equations

Marsden

Raţiu

Scheurle³

2000

Journal of Mathematical Physics

120

176

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Reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of Euler, Jacobi, Lagrange, Hamilton, Routh, Poincaré, and others. The modern vision of mechanics includes, besides the traditional mechanics of particles and rigid bodies, field theories such as electromagnetism, fluid mechanics, plasma physics, solid mechanics as well as quantum mechanics, and relativistic theories, including gravity. Symmetries in these theories vary from obvious translational and rotational symmetries to less obvious particle relabeling symmetries in fluids and plasmas, to subtle symmetries underlying integrable systems. Reduction theory concerns the removal of symmetries and their associated conservation laws. Variational principles, along with symplectic and Poisson geometry, provide fundamental tools for this endeavor. Reduction theory has been extremely useful in a wide variety of areas, from a deeper understanding of many physical theories, including new variational and Poisson structures, to stability theory, integrable systems, as well as geometric phases. This paper surveys progress in selected topics in reduction theory, especially those of the last few decades as well as presenting new results on non-Abelian Routh reduction. We develop the geometry of the associated Lagrange-Routh equations in some detail. The paper puts the new results in the general context of reduction theory and discusses some future directions.

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mentioning

confidence: 99%

Reduction theory and the Lagrange–Routh equations

Marsden

Raţiu

Scheurle³

2000

Journal of Mathematical Physics

120

176

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“…Later he and Prato [11] tackled the case of nonabelian actions (the argument in this paper is different from theirs and is an expansion of the remarks in [6]). The present paper is a continuation of [6] which contained the corresponding results for the case of manifolds.…”

Section: Q(m ω) := Kernel ∂ / − Cokernel ∂ /mentioning

confidence: 92%

Quantization of symplectic orbifolds and group actions

Silva¹,

Guillemin²

1999

Northern California Symplectic Geometry Seminar

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“…We show that the orbittype decomposition of the action of SU(3) C G 2 on M has three components. The moment polytope A can be computed using, for instance, a formula of Guillemin and Prato [11], and is pictured as the shaded region in the figure below on the right. A local model for M in a neighborhood of x shows that SU(3) acts generically freely.…”

Section: Lemma 73 the Stabilizer Gp Of Any Point P In 4-l 1 (X) Is mentioning

confidence: 99%

The classification of transversal multiplicity-free group actions

Woodward¹

1996

Ann Glob Anal Geom

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Multiplicity-free Hamiltonian group actions are the symplectic analogs of multiplicity-free representations, that is, representations in which each irreducible appears at most once. The most well-known examples are toric varieties. The purpose of this paper is to show that under certain assumptions multiplicity-free actions whose moment maps are transversal to a Cartan subalgebra are in one-to-one correspondence with a certain collection of convex polytopes. This result generalizes a theorem of Delzant concerning torus actions.

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Heckman, Kostant, and Steinberg formulas for symplectic manifolds

Cited by 31 publications

References 10 publications

Reduction theory and the Lagrange–Routh equations

Reduction theory and the Lagrange–Routh equations

Quantization of symplectic orbifolds and group actions

The classification of transversal multiplicity-free group actions

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