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144
Hamiltonian torus actions on symplectic orbifolds and toric varieties
, 1995
"... Abstract. In the first part of the paper, we build a foundation for further work on Hamiltonian actions on symplectic orbifolds. Most importantly we prove the orbifold versions of the abelian connectedness and convexity theorems. In the second half, we prove that compact symplectic orbifolds with co ..."
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Cited by 76 (5 self)
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Abstract. In the first part of the paper, we build a foundation for further work on Hamiltonian actions on symplectic orbifolds. Most importantly we prove the orbifold versions of the abelian connectedness and convexity theorems. In the second half, we prove that compact symplectic orbifolds with completely integrable torus actions are classified by convex simple rational polytopes with a positive integer attached to each facet and that all such orbifolds are algebraic toric varieties. Contents
Momentum Maps and Classical Relativistic Fields Part I: Covariant Field Theory
, 1997
"... this paper.) In Part I we develop some of the basic theory of classical fields from a spacetime covariant viewpoint. Throughout we restrict attention to first order theories; these are theories whose Lagrangians involve no higher than first derivatives of the fields ..."
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Cited by 75 (4 self)
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this paper.) In Part I we develop some of the basic theory of classical fields from a spacetime covariant viewpoint. Throughout we restrict attention to first order theories; these are theories whose Lagrangians involve no higher than first derivatives of the fields
Proper group actions and symplectic stratified spaces
 PACIFIC J. MATH
, 1997
"... Let (M,ω) be a Hamiltonian Gspace with a momentum map F: M → g ∗. It is wellknown that if α is a regular value of F and G acts freely and properly on the level set F −1 (G · α), then the reduced space Mα: = F −1 (G · α)/G is a symplectic manifold. We show that if the regularity assumptions are dro ..."
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Cited by 45 (7 self)
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Let (M,ω) be a Hamiltonian Gspace with a momentum map F: M → g ∗. It is wellknown that if α is a regular value of F and G acts freely and properly on the level set F −1 (G · α), then the reduced space Mα: = F −1 (G · α)/G is a symplectic manifold. We show that if the regularity assumptions are dropped the space Mα is a union of symplectic manifolds, and that the symplectic manifolds fit together in a nice way. In other words the reduced space is a symplectic stratified space. This extends results known for the Hamiltonian action of compact groups.
Lagrangian reduction and the double spherical pendulum
 ZAMP
, 1993
"... This paper studies the stability and bifurcations of the relative equilibria of the double spherical pendulum, which has the circle as its symmetry group. This example as well as others with nonabelian symmetry groups, such as the rigid body, illustrate some useful general theory about Lagrangian re ..."
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Cited by 40 (20 self)
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This paper studies the stability and bifurcations of the relative equilibria of the double spherical pendulum, which has the circle as its symmetry group. This example as well as others with nonabelian symmetry groups, such as the rigid body, illustrate some useful general theory about Lagrangian reduction. In particular, we establish a satisfactory global theory of Lagrangian reduction that is consistent with the classical local Routh theory for systems with an abelian symmetry group. 1
Quiver varieties and finite dimensional representations of quantum affine algebras
 J. Amer. Math. Soc
"... Abstract. We study finite dimensional representations of the quantum affine algebra Uq(̂g) using geometry of quiver varieties introduced by the author [29, 44, 45]. As an application, we obtain character formulas expressed in terms of intersection cohomologies of quiver varieties. ..."
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Cited by 36 (5 self)
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Abstract. We study finite dimensional representations of the quantum affine algebra Uq(̂g) using geometry of quiver varieties introduced by the author [29, 44, 45]. As an application, we obtain character formulas expressed in terms of intersection cohomologies of quiver varieties.
SYMPLECTIC AND POISSON STRUCTURES OF CERTAIN MODULI SPACES. II. PROJECTIVE REPRESENTATIONS OF COCOMPACT PLANAR DISCRETE GROUPS
, 1994
"... Let G be a Lie group with a biinvariant metric, not necessarily positive definite. It is shown that a certain construction carried out in an earlier paper for the fundamental group of a closed surface may be extended to an arbitrary infinite orientation preserving cocompact planar discrete group of ..."
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Cited by 34 (13 self)
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Let G be a Lie group with a biinvariant metric, not necessarily positive definite. It is shown that a certain construction carried out in an earlier paper for the fundamental group of a closed surface may be extended to an arbitrary infinite orientation preserving cocompact planar discrete group of euclidean or noneuclidean motions π and yields (i) a symplectic structure on a certain smooth manifold M containing the space Hom(π, G) of homomorphisms and, furthermore, (ii) a hamiltonian Gaction on M preserving the symplectic structure together with a momentum mapping in such a way that the reduced space equals the space Rep(π, G) of representations. More generally, the construction also applies to certain spaces of projective representations. For G compact, the resulting spaces of representations inherit structures of stratified symplectic space in such a way that the strata have finite symplectic volume. For example, MehtaSeshadri moduli spaces of semistable holomorphic parabolic bundles with rational weights or spaces closely related to them arise in this way by symplectic reduction in finite dimensions.
The reduced EulerLagrange equations
 Fields Institute Comm
, 1993
"... Marsden and Scheurle [1993] studied Lagrangian reduction in the context of momentum map constraints—here meaning the reduction of the standard EulerLagrange system restricted to a level set of a momentum map. This provides a Lagrangian parallel to the reduction of symplectic manifolds. The present ..."
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Cited by 31 (15 self)
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Marsden and Scheurle [1993] studied Lagrangian reduction in the context of momentum map constraints—here meaning the reduction of the standard EulerLagrange system restricted to a level set of a momentum map. This provides a Lagrangian parallel to the reduction of symplectic manifolds. The present paper studies the Lagrangian parallel of Poisson reduction for Hamiltonian systems. For the reduction of a Lagrangian system on a level set of a conserved quantity, a key object is the Routhian, which is the Lagrangian minus the mechanical connection paired with the fixed value of the momentum map. For unconstrained systems, we use a velocity shifted Lagrangian, which plays the role of the Routhian in the constrained theory. Hamilton’s variational principle for the EulerLagrange equations breaks up into two sets of equations that represent a set of EulerLagrange equations with gyroscopic forcing that can be written in terms of the curvature of the connection for horizontal variations, and into the EulerPoincaré equations for the vertical variations. This new set of equations is what we call the reduced EulerLagrange equations, and it includes the EulerPoincaré and the Hamel equations as special cases. We illustrate this methodology for a rigid body with internal rotors and for a particle moving in a magnetic field. 1
Stability and drift of underwater vehicle dynamics: mechanical systems with rigid motion symmetry
 Physica D
, 1997
"... This paper develops the stability theory of relative equilibria for mechanical systems with symmetry. It is especially concerned with systems that have a noncompact symmetry group, such as the group of Euclidean motions, and with relative equilibria for such symmetry groups. For these systems with r ..."
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Cited by 26 (7 self)
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This paper develops the stability theory of relative equilibria for mechanical systems with symmetry. It is especially concerned with systems that have a noncompact symmetry group, such as the group of Euclidean motions, and with relative equilibria for such symmetry groups. For these systems with rigid motion symmetry, one gets stability but possibly with drift in certain rotational as well as translational directions. Motivated by questions on stability of underwater vehicle dynamics, it is of particular interest that, in some cases, we can allow the relative equilibria to have nongeneric values of their momentum. The results are proved by combining theorems of Patrick with the technique of reduction by stages. This theory is then applied to underwater vehicle dynamics. The stability of specific relative equilibria for the underwater vehicle is studied. For example, we find conditions for Liapunov stability of the steadily rising and possibly spinning, bottomheavy vehicle, which corresponds to a relative equilibrium with nongeneric momentum. The results of this paper should prove
Geometric mechanics, Lagrangian reduction and nonholonomic systems
 in Mathematics Unlimited2001 and Beyond
, 2001
"... This paper surveys selected recent progress in geometric mechanics, focussing on Lagrangian reduction and gives some new applications to nonholonomic systems, that is, mechanical systems with constraints typified by rolling without slipping. Reduction theory for mechanical systems with symmetry has ..."
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Cited by 22 (5 self)
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This paper surveys selected recent progress in geometric mechanics, focussing on Lagrangian reduction and gives some new applications to nonholonomic systems, that is, mechanical systems with constraints typified by rolling without slipping. 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 mechanics ranges 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 utilizing their associated conservation laws. Reduction theory has been extremely useful in a wide variety of areas, from a deeper understanding of many
Singular reduction and quantization
, 1996
"... Abstract. Consider a compact prequantizable symplectic manifold M on which a compact Lie group G acts in a Hamiltonian fashion. The “quantization commutes with reduction ” theorem asserts that the Ginvariant part of the equivariant index of M is equal to the RiemannRoch number of the symplectic qu ..."
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Cited by 22 (2 self)
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Abstract. Consider a compact prequantizable symplectic manifold M on which a compact Lie group G acts in a Hamiltonian fashion. The “quantization commutes with reduction ” theorem asserts that the Ginvariant part of the equivariant index of M is equal to the RiemannRoch number of the symplectic quotient of M, provided the quotient is nonsingular. We extend this result to singular symplectic quotients, using partial desingularizations of the symplectic quotient to define its RiemannRoch number. By similar methods we also compute multiplicities for the equivariant index of the dual of a prequantum bundle, and furthermore show that the arithmetic genus of a Hamiltonian Gmanifold is invariant under symplectic reduction.