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DIRAC ACTIONS AND LU’S LIE ALGEBROID

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Abstract

Poisson actions of Poisson Lie groups have an interesting and rich geometric structure. We will generalize some of this structure to Dirac actions of Dirac Lie groups. Among other things, we extend a result of Jiang-Hua Lu, which states that the cotangent Lie algebroid and the action algebroid for a Poisson action form a matched pair. We also give a full classification of Dirac actions for which the base manifold is a homogeneous space H/K, obtaining a generalization of Drinfeld’s classification for the Poisson Lie group case.

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References

  1. A. Alekseev, H. Bursztyn, E. Meinrenken, Pure spinors on Lie groups, Astérisque 327 (2009), 131–199.

    MATH  MathSciNet  Google Scholar 

  2. A. Alekseev, A. Malkin, E. Meinrenken, Lie group valued moment maps, J. Differential Geom. 48 (1998), no. 3, 445–495.

    Article  MATH  MathSciNet  Google Scholar 

  3. A. Alekseev, P. Xu, Derived brackets and Courant algebroids, unfinished manuscript (2002).

  4. C. Blohmann, A. Weinstein, Group-like objects in Poisson geometry and algebra, in: Poisson Geometry in Mathematics and Physics, Contemp. Math., Vol. 450, Amer. Math. Soc., Providence, RI, 2008, pp. 25–39.

  5. H. Bursztyn, A. Cabrera, M. del Hoyo, Vector bundles over Lie groupoids and algebroids, preprint, arXiv:1410.5135 (2014).

  6. H. Bursztyn, G. Cavalcanti, M. Gualtieri, Reduction of Courant algebroids and generalized complex structures, Adv. Math. 211 (2007), no. 2, 726–765.

    Article  MATH  MathSciNet  Google Scholar 

  7. H. Bursztyn, M. Crainic, Dirac structures, momentum maps, and quasi-Poisson manifolds, in: The Breadth of Symplectic and Poisson Geometry, Progr. Math., Vol. 232, Birkhäuser Boston, Boston, MA, 2005, pp. 1–40.

  8. H. Bursztyn, M. Crainic, Dirac geometry, quasi-Poisson actions and D/G-valued moment maps, J. Differential Geom. 82 (2009), no. 3, 501–566.

    Article  MATH  MathSciNet  Google Scholar 

  9. H. Bursztyn, M. Crainic, P. Ševera, Quasi-Poisson structures as Dirac structures, Travaux mathématiques, Fasc. XVI, Trav. Math., XVI, Univ. Luxemb., Luxembourg, 2005, pp. 41–52.

  10. H. Bursztyn, D. Iglesias Ponte, P. Severa, Courant morphisms and moment maps, Math. Res. Lett. 16 (2009), no. 2, 215–232.

    Article  MATH  MathSciNet  Google Scholar 

  11. T. Courant, Dirac manifolds, Trans. Amer. Math. Soc. 319 (1990), no. 2, 631–661.

    Article  MATH  MathSciNet  Google Scholar 

  12. T. Courant, A. Weinstein, Beyond Poisson structures, in: Action Hamiltoniennes de Groupes. Troisième Théorème de Lie (Lyon, 1986), Travaux en Cours, Vol. 27, Hermann, Paris, 1988, pp. 39–49.

  13. P. Delorme, Classification des triples de Manin pour les algèbres de Lie réductives complexes, J. Algebra 246 (2001), no. 1, 97–174, with an appendix by G. Macey.

  14. I. Ya. Dorfman, Dirac structures and integrability of nonlinear evolution equations, Nonlinear Science - theory and applications, Wiley, Chichester, 1993.

  15. V. G. Drinfeld, Quantum groups, in: Proceedings of the International Congress of Mathematicians, Vol. 1, 2, Berkeley, Calif., 1986, Amer. Math. Soc., Providence, RI, 1987, pp. 798–820.

  16. S. Evens, J.-H. Lu, On the variety of Lagrangian subalgebras. I, Ann. Sci. École Norm. Sup. (4) 34 (2001), no. 5, 631–668.

  17. S. Evens, J.-H. Lu, On the variety of Lagrangian subalgebras. II, Ann. Sci. École Norm. Sup. (4) 39 (2006), no. 2, 347–379.

  18. R. Fernandes, D. Iglesias Ponte, Integrability of Poisson-Lie group actions, Lett. Math. Phys. 90 (2009), no. 1–3, 137–159.

    Article  MATH  MathSciNet  Google Scholar 

  19. H. Flaschka, T. Ratiu, A convexity theorem for Poisson actions of compact Lie groups, Ann. Sci. Ecole Norm. Sup. 29 (1996), no. 6, 787–809.

    Article  MATH  MathSciNet  Google Scholar 

  20. A. Gracia-Saz, R. Mehta, VB-groupoids and representation theory of Lie groupoids, arXiv:1007.3658v6 (2016), to appear in J. Sympl. Geom.

  21. M. Grützmann, M. Stiénon, Matched pairs of Courant algebroids, Indag. Math. (N.S.) 25 (2014), no. 5, 977–991.

  22. M. Jotz, Dirac groupoids and Dirac bialgebroids, arXiv:1403.2934v2 (2015).

  23. M. Jotz, Dirac Lie groups, Dirac homogeneous spaces and the theorem of Drinfeld, Indiana Univ. Math. J. 60 (2011), no. 1, 319–366.

    Article  MATH  MathSciNet  Google Scholar 

  24. E. Karolinsky, A classification of Poisson homogeneous spaces of complex reductive Poisson-Lie groups, in: Poisson Geometry (Warsaw, 1998), Banach Center Publ., Vol. 51, Polish Acad. Sci., Warsaw, 2000, pp. 103–108.

  25. E. Karolinsky, S. Lyapina, Lagrangian subalgebras in g × g, where \( \mathfrak{g} \) is a real simple Lie algebra of real rank one, Travaux mathématiques. Fasc. XVI, Trav. Math., XVI, Univ. Luxemb., Luxembourg, 2005, pp. 229–236.

  26. Y. Kosmann-Schwarzbach, F. Magri, Poisson-Lie groups and complete integrability. I. Drinfel’d bialgebras, dual extensions and their canonical representations, Ann. Inst. H. Poincaré Phys. Théor. 49 (1988), no. 4, 433–460.

  27. D. Li-Bland, E. Meinrenken, Courant algebroids and Poisson geometry, Internat. Math. Research Notices 11 (2009), 2106–2145.

    MATH  MathSciNet  Google Scholar 

  28. D. Li-Bland, E. Meinrenken, Dirac Lie groups, Asian J. Math. 18 (2014), no. 5, 779–816.

    Article  MATH  MathSciNet  Google Scholar 

  29. D. Li-Bland, P. Ševera, Quasi-Hamiltonian groupoids and multiplicative Manin pairs, Internat. Math. Research Notices 2011 (2011), 2295–2350.

    MATH  MathSciNet  Google Scholar 

  30. Z.-J. Liu, A. Weinstein, P. Xu, Manin triples for Lie bialgebroids, J. Differential Geom. 45 (1997), no. 3, 547–574.

    Article  MATH  MathSciNet  Google Scholar 

  31. J.-H. Lu, Momentum mappings and reduction of Poisson actions, in: Symplectic Geometry, Groupoids, and Integrable Systems (Berkeley, CA, 1989), Springer, New York, 1991, pp. 209–226.

  32. J.-H. Lu, A note on Poisson homogeneous spaces, in: Poisson Geometry in Mathematics and Physics, Contemp. Math., Vol. 450, Amer. Math. Soc., Providence, RI, 2008, pp. 173–198.

  33. J.-H. Lu, A. Weinstein, Poisson Lie groups, dressing transformations, and Bruhat decompositions, J. Differential Geom. 31 (1990), no. 2, 501–526.

    Article  MATH  MathSciNet  Google Scholar 

  34. J.-H. Lu, Poisson homogeneous spaces and Lie algebroids associated to Poisson actions , Duke Math. J. 86 (1997), no. 2, 261–304.

    Article  MATH  MathSciNet  Google Scholar 

  35. K. Mackenzie, Double Lie algebroids and second-order geometry. I, Adv. Math. 94 (1992), no. 2, 180–239.

    Article  MATH  MathSciNet  Google Scholar 

  36. K. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Mathematical Society Lecture Note Series, Vol. 213, Cambridge University Press, Cambridge, 2005.

  37. J. Marrero, E. Padron, M. Rodriguez-Olmos, Reduction of a symplectic-like Lie algebroid with momentum map and its application to fiberwise linear Poisson structures, J. Physics A: Math. Theor. 45 (2012), 165–201.

    Article  MATH  MathSciNet  Google Scholar 

  38. B. Milburn, Two categories of Dirac manifolds, arXiv:0712.2636 (2007).

  39. T. Mokri, Matched pairs of Lie algebroids, Glasgow Math. J. 39 (1997), no. 2, 167–181.

    Article  MATH  MathSciNet  Google Scholar 

  40. C. Ortiz, Multiplicative Dirac structures on Lie groups, C. R. Math. Acad. Sci. Paris 346 (2008), no. 23–24, 1279–1282.

    Article  MATH  MathSciNet  Google Scholar 

  41. J. Pradines, Remarque sur le groupoïde cotangent de Weinstein–Dazord, C. R. Acad. Sci. Paris Sér. I Math. 306 (1988), no. 13, 557–560.

    MATH  MathSciNet  Google Scholar 

  42. P. Robinson, The Classification of Dirac Homogeneous Spaces, Thesis, University of Toronto, 2014 arXiv:1411.2958 (2014).

  43. M. A. Semenov–Tian-Shansky, Dressing transformations and Poisson group actions, Publ. Res. Inst. Math. Sci. 21 (1985), no. 6, 1237–1260.

  44. P. Ševera, Letters to Alan Weinstein, http://sophia.dtp.fmph.uniba.sk/~severa/letters/, 1998–2000.

  45. P. Ševera, F. Valach, Lie groups in quasi-Poisson geometry and braided Hopf algebras , arXiv:1604.07164 (2016).

  46. L. Stefanini, On morphic actions and integrability of LA-groupoids, PhD thesis, Zürich 2008, arxiv.org/abs/0902.2228 (2009).

  47. K. Uchino, Remarks on the definition of a Courant algebroid, Lett. Math. Phys. 60 (2002), no. 2, 171–175.

    Article  MATH  MathSciNet  Google Scholar 

  48. P. Xu, On Poisson groupoids, Internat. J. Math. 6 (1995), no. 1, 101–124.

    Article  MATH  MathSciNet  Google Scholar 

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  1. Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario, M4S2E4, Canada

    E. MEINRENKEN

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MEINRENKEN, E. DIRAC ACTIONS AND LU’S LIE ALGEBROID. Transformation Groups 22, 1081–1124 (2017). https://doi.org/10.1007/s00031-017-9424-y

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