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Communications in Analysis and Mechanics

2024, Volume 16, Issue 3: 448-456. doi: 10.3934/cam.2024021
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No-go theorems for r-matrices in symplectic geometry

  • Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Ernst-Zermelo-Str. 1, 79104 Freiburg i. Br., Germany
  • Received: 17 November 2023 Revised: 31 January 2024 Accepted: 16 May 2024 Published: 01 July 2024

  • 53D05, 16W25

  • If a triangular Lie algebra acts on a smooth manifold, it induces a Poisson bracket on it. In case this Poisson structure is actually symplectic, we show that this already implies the existence of a flat connection on any vector bundle over the manifold the Lie algebra acts on, in particular the tangent bundle. This implies, among other things, that CPn and higher genus Pretzel surfaces cannot carry symplectic structures that are induced by triangular Lie algebras.

    Citation: Jonas Schnitzer. No-go theorems for r-matrices in symplectic geometry[J]. Communications in Analysis and Mechanics, 2024, 16(3): 448-456. doi: 10.3934/cam.2024021

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  • If a triangular Lie algebra acts on a smooth manifold, it induces a Poisson bracket on it. In case this Poisson structure is actually symplectic, we show that this already implies the existence of a flat connection on any vector bundle over the manifold the Lie algebra acts on, in particular the tangent bundle. This implies, among other things, that CPn and higher genus Pretzel surfaces cannot carry symplectic structures that are induced by triangular Lie algebras.




    [1] F. Bayen, M. Flato, C. Frønsdal, A. Lichnerowicz, D. Sternheimer, Deformation theory and quantization. I. Deformations of symplectic structures, Ann. Phys., 111 (1978), 61–151. https://doi.org/10.1016/0003-4916(78)90224-5 doi: 10.1016/0003-4916(78)90224-5
    [2] M. Gerstenhaber, On the Deformation of Rings and Algebras, Ann. Math., 79 (1964), 59–103. https://doi.org/10.2307/1970484 doi: 10.2307/1970484
    [3] M. Kontsevich, Deformation quantization of {P}oisson manifolds, Lett. Math. Phys., 66 (2003), 157–216. https://doi.org/10.1023/B:MATH.0000027508.00421.bf doi: 10.1023/B:MATH.0000027508.00421.bf
    [4] V. Dolgushev, Covariant and equivariant formality theorems, Adv. Math., 191 (2005), 147–177. https://doi.org/10.1016/j.aim.2004.02.001 doi: 10.1016/j.aim.2004.02.001
    [5] C. Kassel, Quantum Groups, Graduate Texts in Mathematics, Springer-Verlag, 1995.
    [6] C. Esposito, J. Schnitzer, S. Waldmann, A universal construction of universal deformation formulas, Drinfeld twists and their positivity, Pacific J. Math., 291 (2017), 319–358. https://doi.org/10.2140/pjm.2017.291.319 doi: 10.2140/pjm.2017.291.319
    [7] P. Bieliavsky, C. Esposito, S. Waldmann, T. Weber, Obstructions for twist star products, Lett. Math. Phys., 108 (2018), 1341–1350. https://doi.org/10.1007/s11005-017-1034-z doi: 10.1007/s11005-017-1034-z
    [8] F. D'Andrea, T. Weber, Twist star products and Morita equivalence, C. R. Math., 355 (2017), 1178–1184. https://doi.org/10.1016/j.crma.2017.10.012 doi: 10.1016/j.crma.2017.10.012
    [9] V. Drinfel'd, Constant quasiclassical solutions of the Yang–Baxter quantum equation, Dokl. Akad. Nauk SSSR, 273 (1983), 531–535. In Russian; translated in Soviet Math. Dokl. 28 (1983), 667–671.
    [10] O. Baues, V. Cortés, Symplectic Lie groups, Astérisque, 379 (2016).
    [11] L. P. Castellanos Moscoso, H. Tamaru, A classification of left-invariant symplectic structures on some Lie groups, Beitr. Algebra Geom., 64 (2023), 471–491. https://doi.org/10.1007/s13366-022-00643-1 doi: 10.1007/s13366-022-00643-1
    [12] G. Ovando, Four dimensional symplectic Lie algebras, Beitr. Algebra Geom., 47 (2006), 419–434.
    [13] S. Salamon, Complex structures on nilpotent Lie algebras, J. Pure Appl. Algebra, 157 (2001), 311–333. https://doi.org/10.1016/S0022-4049(00)00033-5 doi: 10.1016/S0022-4049(00)00033-5
    [14] D. V. Alekseevsky, A. M. Perelomov, Poisson and symplectic structures on Lie algebras. I, J. Geom. Phys., 22 (1997), 191–211. https://doi.org/10.1016/S0393-0440(96)00025-3 doi: 10.1016/S0393-0440(96)00025-3
    [15] P. Etingof, O. Schiffmann, Lectures on Quantum groups, International Press, Boston, 1998.
    [16] V. Drinfel'd, On Poisson homogeneous spaces of Poisson-Lie groups, Theor. Math. Phys., 95 (1993), 524–525. https://doi.org/10.1007/BF01017137 doi: 10.1007/BF01017137
    [17] J. H. Lu, A. Weinstein, Poisson Lie groups, dressing transformations, and Bruhat Decompositions, J. Diff. Geom., 31 (1990), 501–526. https://doi.org/10.4310/jdg/1214444324
    [18] R. Nest, B. Tsygan, Deformations of symplectic Lie algebroids, deformations of holomorphic symplectic structures, and index theorems, Asian J. Math., 5 (2001), 599–635. https://dx.doi.org/10.4310/AJM.2001.v5.n4.a2 doi: 10.4310/AJM.2001.v5.n4.a2
    [19] J. Milnor, On the existence of a connection with curvature zero, Comment. Math. Helv., 32 (1958), 215–223. https://doi.org/10.1007/BF02564579 doi: 10.1007/BF02564579

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