Abstract
We investigate the fine structure of the symplectic foliations of Poisson homogeneous spaces. Two general results are proved for weak splittings of surjective Poisson submersions from Heisenberg and Drinfeld doubles. The implications of these results are that the torus orbits of symplectic leaves of the quotients can be explicitly realized as Poisson–Dirac submanifolds of the torus orbits of the doubles. The results have a wide range of applications to many families of real and complex Poisson structures on flag varieties. Their torus orbits of leaves recover important families of varieties such as the open Richardson varieties.
The author was supported in part by NSF grants DMS-1001632 and DMS-1303036.
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References
A. A. Belavin and V. G. Drinfeld, Triangular equations and simple Lie algebras, Math. Phys. Rev. 4 (1984), 93–165.
A. Berenstein, S. Fomin, and A. Zelevinsky, Cluster algebras III, Upper bounds and double Bruhat cells, Duke Math. J. 126 (2005), 1–52.
A. Berenstein and J. Greenstein, Quantum folding, Int. Math. Res. Not. IMRN 2011, no. 21, 4821–4883.
S. Billey and I. Coskun, Singularities of generalized Richardson varieties, Comm. Algebra 40 (2012), 1466–1495.
N. Chevalier, Algèbres amassées et positivité totale, Ph.D. thesis, Univ. of Caen, 2012.
M. Crainic and R. L. Fernandes, Integrability of Poisson brackets, J. Diff. Geom. 66 (2004), 71–137.
C. De Concini and C. Procesi, Complete symmetric varieties, in: Invariant theory (Montecatini, 1982), Lect. Notes Math. 996 (1983), 1–44.
P. Delorme, Classification des triples de Manin pour les algèbres de Lie réductives complex, with an appendix by G. Macey, J. Algebra 246 (2001), 97–174.
V. G. Drinfeld, On Poisson homogeneous spaces of Poisson–Lie groups, Theor. and Math. Phys. 95 (1993), 524–525.
S. Evens and J.-H. Lu, On the variety of Lagrangian subalgebras. I, Ann. Sci. École Norm. Sup. (4), 34 (2001), 631–668.
S. Evens and J.-H. Lu, Poisson geometry of the Grothendieck resolution of a complex semisimple Lie group, Mosc. Math. J. 7 (2007), 613–642.
Ph. Foth and J.-H. Lu, Poisson structures on complex flag manifolds associated with real forms, Trans. Amer. Math. Soc. 358 (2006), 1705–1714.
T. J. Hodges and T. Levasseur, Primitive ideals of \(\mathbb{C}_{q}[\mathrm{SL}(3)]\), Comm. Math. Phys. 156 (1993), 581–605.
M. Gekhtman, M. Shapiro, and A. Vainshtein, Cluster algebras and Poisson geometry, Mosc. Math. J. 3 (2003), 899–934
M. Gekhtman, M. Shapiro, and A. Vainshtein, Exotic cluster structures onSLn : the Cremmer–Gervais case, preprint arXiv:1307.1020.
K. R. Goodearl and M. Yakimov, Poisson structures on affine spaces and flag varieties. II, Trans. Amer. Math. Soc. 361 (2009), 5753–5780.
A. Joseph, On the prime and primitive spectra of the algebra of functions on a quantum group, J. Algebra 169 (1994), 441–511.
A. W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Progress in Math. 140, Birkhäuser, Cambridge, 2002.
A. Knutson, T. Lam, and D. E. Speyer, Projections of Richardson varieties, J. Reine Angew. Math. 687 (2014), 133–157.
M. Kogan and A. Zelevinsky, On symplectic leaves and integrable systems in standard complex semisimple Poisson-Lie groups, Int. Math. Res. Not. 2002, no. 32, 1685–1702.
B. Leclerc, Cluster structures on strata of flag varieties, preprint arXiv:1402.4435.
J.-H. Lu and M. Yakimov, Group orbits and regular partitions of Poisson manifolds, Comm. Math. Phys. 283 (2008), 729–748.
G. Muller and D. E. Speyer, Cluster algebras of Grassmannians are locally acyclic, preprint arXiv:1401.5137.
A. Postnikov, Total positivity, Grassmannians, and networks, preprint arXiv:math/0609764.
K. Rietsch and L. Williams, The totally nonnegative part of G∕P is a CW complex, Transform. Groups 13 (2008), 839–853.
I. Vaisman, Dirac submanifolds of Jacobi manifolds, in: The breadth of symplectic and Poisson geometry, 603–622, Progr. Math. 232, Birkhäuser Boston, Cambridge, MA, 2005.
B. Webster and M. Yakimov, A Deodhar-type stratification on the double flag variety, Transform. Groups 12 (2007), 769–785.
P. Xu, Dirac submanifolds and Poisson involutions, Ann. Sci. École Norm. Sup. (4) 36 (2003), 403–430.
M. Yakimov, Symplectic leaves of complex reductive Poisson–Lie groups, Duke Math. J. 112 (2002), 453–509.
M. Yakimov, A classification of H-primes of quantum partial flag varieties, Proc. Amer. Math. Soc. 138 (2010), 1249–1261.
Acknowledgements
I would like to thank Bernard Leclerc and Jiang-Hua Lu for their valuable comments and suggestions on the first version of the preprint. I am also grateful to I. Dimitrov, G. Mason, S. Montgomery, I. Penkov, V.S. Varadarajan, and J. Wolf for the opportunity to present these and related results at the Lie theory meetings at UCB, UCLA, UCSC, and USC.
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Yakimov, M. (2014). Weak Splittings of Quotients of Drinfeld and Heisenberg Doubles. In: Mason, G., Penkov, I., Wolf, J. (eds) Developments and Retrospectives in Lie Theory. Developments in Mathematics, vol 37. Springer, Cham. https://doi.org/10.1007/978-3-319-09934-7_9
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