Convexity Properties of the Moment Mapping Re-examined

Sjamaar, Reyer

doi:10.1006/aima.1998.1739

Cited by 101 publications

(132 citation statements)

References 27 publications

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“…In fact,Γ(s) consists of the rational points of the image of Γ(s, K) (cf., e.g., [Sj,Theorem 7.6]). Since x i corresponds with 2ω i / α i , α i , the theorem follows from Theorem 21.…”

Section: L-movability and The Eigenvalue Problemmentioning

confidence: 99%

Eigenvalue problem and a new product in cohomology of flag varieties

2006

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“…In fact,Γ(s) consists of the rational points of the image of Γ(s, K) (cf., e.g., [Sj,Theorem 7.6]). Since x i corresponds with 2ω i / α i , α i , the theorem follows from Theorem 21.…”

Section: L-movability and The Eigenvalue Problemmentioning

confidence: 99%

Eigenvalue problem and a new product in cohomology of flag varieties

2006

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“…Sjamaar [35] has given another proof of the convexity theorems, based on ideas coming from Kähler and algebraic geometry. His proof, even though it gives the most complete information on these polytopes, uses strongly the symplectic form and it is not known how to generalize his technique to other manifolds and other types of actions, such as Poisson actions of Poisson-Lie groups.…”

Section: Theorem (Atiyah-guillemin-sternberg) Let M Be a Compact Conmentioning

confidence: 99%

Openness and convexity for momentum maps

BIRTEA

Ortega

Raţiu

2008

Trans. Amer. Math. Soc.

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Abstract. The purpose of this paper is finding the essential attributes underlying the convexity theorems for momentum maps. It is shown that they are of a topological nature; more specifically, we show that convexity follows if the map is open onto its image and has the so-called local convexity data property. These conditions are satisfied in all the classical convexity theorems and hence they can, in principle, be obtained as corollaries of a more general theorem that has only these two hypotheses. We also prove a generalization of the socalled Local-to-Global Principle that only requires the map to be closed and to have a normal topological space as domain, instead of using a properness condition. This allows us to generalize the Flaschka-Ratiu convexity theorem to noncompact manifolds.

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“…For the compact case, see [At], [GS1], and [Sj,Theorem 6.5]. For proper moment maps to open convex sets and a brief history, see [LMTW].…”

Section: Stability: As a Map To φ(M ) The Moment Map Is Openmentioning

confidence: 99%

Centered complexity one Hamiltonian torus actions

Karshon

Tolman²

2001

Trans. Amer. Math. Soc.

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Abstract. We consider symplectic manifolds with Hamiltonian torus actions which are "almost but not quite completely integrable": the dimension of the torus is one less than half the dimension of the manifold. We provide a complete set of invariants for such spaces when they are "centered" and the moment map is proper. In particular, this classifies the preimages under the moment map of all sufficiently small open sets, which is an important step towards global classification. As an application, we construct a full packing of each of the Grassmannians Gr + (2, R 5 ) and Gr + (2, R 6 ) by two equal symplectic balls.

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Convexity Properties of the Moment Mapping Re-examined

Cited by 101 publications

References 27 publications

Eigenvalue problem and a new product in cohomology of flag varieties

Eigenvalue problem and a new product in cohomology of flag varieties

Openness and convexity for momentum maps

Centered complexity one Hamiltonian torus actions

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