Reduction of Poisson algebras at nonzero momentum values |
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| Affiliation: | 1. Food Science and Technology Program, Beijing Normal University-Hong Kong Baptist University United International College, Zhuhai 519087, China;2. Hebei Normal University of Science and Technology, Qinhuangdao, Hebei 066600, China;3. Kazakhstan Medical University “KSPH”, Almaty 050060, Kazakhstan;1. Dept. of Electrical Engineering, Universidad Politécnica de Cartagena, 30202 Cartagena, Spain;2. Dept. of Hydraulic, Energy and Environmental Engineering, Universidad Politécnica de Madrid, 28040, Spain;3. Dept. of Civil Engineering, Universidad Politécnica de Cartagena, 30202 Cartagena, Spain;1. Department of Mathematics, University of Geneva, 7-9 rue du Conseil Général, c.p. 64, 1211 Geneva 4, Switzerland;2. Hamilton Mathematics Institute and School of Mathematics, Trinity College Dublin, Dublin 2, Ireland;3. Simons Center for Geometry and Physics, Stony Brook, USA |
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| Abstract: | ![]()
A reduction of a Poisson manifold using the ideal I(J) generated by the momentum map was introduced by Śniatycki and Weinstein (1983). This reduction has been extended to nonzero momentum values μ by two methods: by shifting to zero momentum on a larger space, the product with the coadjoint orbit; and by the method of Wilbour and Kimura (1991, 1993) using the modified ideal I(J − μ). It is shown that these two methods produce isomorphic reduced algebras under the assumptions that the symmetry group is connected and that the stabilizer group of μ also is connected. If the latter assumption fails, the shifted reduced algebra is isomorphic to a (possibly proper) subalgebra of the Wilbour-Kimura algebra. |
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