Abstract
Given any Poisson action G×P→P of a Poisson–Lie group G we construct an object Ω=T *G*T* P which has both a Lie groupoid structure and a Lie algebroid structure and which is a half-integrated form of the matched pair of Lie algebroids which J.-H. Lu associated to a Poisson action in her development of Drinfeld's classification of Poisson homogeneous spaces. We use Ω to give a general reduction procedure for Poisson group actions, which applies in cases where a moment map in the usual sense does not exist. The same method may be applied to actions of symplectic groupoids and, most generally, to actions of Poisson groupoids.
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References
Coste, A., Dazord, P. and Weinstein, A.: Groupoïdes symplectiques, Publ. Dép. Math. Univ. Lyon, I, 2/A-1987, 1–65.
Cannas da Silva, A. and Weinstein, A: GeometricModels for Noncommutative Algebras, Berkeley Math Lecture Notes 10, Amer. Math. Soc., Providence, 1999.
Drinfel'd, V. G.: On Poisson homogeneous spaces of Poisson-Lie groups, Theoret. Math. Phys. 95(2) (1993), 226–227.
Higgins, P. J. and Mackenzie, K. C. H.: Algebraic constructions in the category of Lie algebroids, J. Algebra 129 (1990), 194–230.
Huebschmann. J.: Poisson cohomology and quantization, J. Reine Angew. Math. 408 (1990), 57–113.
Karasev, M. V.: The Maslov quantization conditions in higher cohomology and analogs of notions developed in Lie theory for canonical fibre bundles of symplectic manifolds, I, II, Selecta Math. Soviet 8 (1989), 213–234, 235-258. Preprint, Moscov. Inst. Electron Mashinostroeniya, 1981, deposited at VINITI, 1982.
Karasev, M. V.: Analogues of objects of the theory of Lie groups for nonlinear Poisson brackets, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), 508–538, 638. English transl. Math. USSR-Izv. 28(3) (1987), 497-527.
Lu, Jiang-Hua: Momentum mappings and reduction of Poisson actions, In: Symplectic Geometry, Groupoids, and Integrable Systems (Berkeley, CA, 1989), Springer, New York, 1991, pp. 209–226.
Lu, Jiang-Hua: Poisson homogeneous spaces and Lie algebroids associated to Poisson actions, Duke Math. J. 86 (1997), 261–304.
Lu, Jiang-Hua and Weinstein, A.: Groupo ïdes symplectiques doubles des groupes de Lie-Poisson, C.R. Acad. Sci. Paris Sér. I Math. 309 (1989).
Lu, Jiang-Hua and Weinstein, A.: Poisson Lie groups, dressing transformations, and Bruhat decompositions, J. Differential Geom. 31 (1990), 501-526.
Mackenzie, K.: Lie groupoids and Lie algebroids in differential geometry, London Math. Soc. Lecture Note Ser. 124, Cambridge Univ. Press, 1987.
Mackenzie, K. C. H.: Double Lie algebroids and second-order geometry, I. Adv. Math. 94(2) (1992), 180–239.
Mackenzie, K. C. H.: Lie algebroids and Lie pseudoalgebras, Bull. London Math. Soc. 27(2) (1995), 97–147.
Mackenzie, K. C. H.: Drinfel'd doubles and Ehresmann doubles for Lie algebroids and Lie bialgebroids, Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 74–87.
Mackenzie, K. C. H.: Double Lie algebroids and second-order geometry, II, Adv. Math. 154 (2000), 46–75.
Mackenzie, K. C. H.: Notions of double for Lie algebroids, submitted.
Mackenzie, K. C. H. and Xu, Ping: Lie bialgebroids and Poisson groupoids, Duke Math. J. 73(2) (1994), 415–452.
Mackenzie, K. C. H. and Xu, Ping: Classical lifting processes and multiplicative vector fields, Quart. J. Math. Oxford (2) 49 (1998), 59–85.
Marsden, J. E. and Raţiu, T.: Reduction of Poisson manifolds, Lett. Math. Phys. 11(2) (1986), 161–169.
Mikami, K. and Weinstein, A.: Moments and reduction for symplectic groupoids, Publ. Res. Inst. Math. Sci. 24 (1988), 121–140.
Mokri, T.: Matched pairs of Lie algebroids, Glasgow Math. J. 39 (1997), 167–181.
Vaisman, I.: Lectures on the Geometry of Poisson Manifolds, Progr. Math., Birkhüuser, Basel, 1994.
Weinstein, A.: The local structure of Poisson manifolds, J. Differential Geom. 18 (1983), 523–557, Errata and Addenda, J. Differential Geom. 22 (1985), 255.
Weinstein, A.: Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc. (N.S.), 16 (1987), 101–104.
Weinstein, A.: Coisotropic calculus and Poisson groupoids, J. Math. Soc. Japan 40 (1988), 705–727.
Weinstein, A.: Some remarks on dressing transformations, J. Fac. Sci. Univ. Tokyo. Sect. 1A, Math. 36 (1988), 163–167.
Weinstein, A.: Affine Poisson structures, Internat. J. Math. 1 (1990), 343–360.
Xu, Ping: Symplectic groupoids of reduced Poisson spaces, C.R. Acad. Sci. Paris Sèr. I Math. 314 (1992), 457–461.
Xu, Ping: On Poisson groupoids, Internat. J. Math. 6(1) (1995), 101–124.
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Mackenzie, K.C.H. A Unified Approach to Poisson Reduction. Letters in Mathematical Physics 53, 215–232 (2000). https://doi.org/10.1023/A:1011055510672
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DOI: https://doi.org/10.1023/A:1011055510672