Communications in Mathematical Physics

, Volume 322, Issue 1, pp 49–71 | Cite as

Poisson Algebras of Block-Upper-Triangular Bilinear Forms and Braid Group Action

  • Leonid Chekhov
  • Marta MazzoccoEmail author


In this paper we study a quadratic Poisson algebra structure on the space of bilinear forms on \({\mathbb{C}^{N}}\) with the property that for any \({n, m \in \mathbb{N}}\) such that n mN, the restriction of the Poisson algebra to the space of bilinear forms with a block-upper-triangular matrix composed from blocks of size \({m \times m}\) is Poisson. We classify all central elements and characterise the Lie algebroid structure compatible with the Poisson algebra. We integrate this algebroid obtaining the corresponding groupoid of morphisms of block-upper-triangular bilinear forms. The groupoid elements automatically preserve the Poisson algebra. We then obtain the braid group action on the Poisson algebra as elementary generators within the groupoid. We discuss the affinisation and quantisation of this Poisson algebra, showing that in the case m = 1 the quantum affine algebra is the twisted q-Yangian for \({\mathfrak{o}_{n}}\) and for m = 2 is the twisted q-Yangian for \({(\mathfrak{sp}_{2n})}\). We describe the quantum braid group action in these two examples and conjecture the form of this action for any m > 2. Finally, we give an R-matrix interpretation of our results and discuss the relation with Poisson–Lie groups.


Bilinear Form Poisson Bracket Central Element Poisson Structure Braid Group 
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  1. 1.
    Bondal A.: A symplectic groupoid of triangular bilinear forms and the braid groups. Izv. Math. 68, 659–708 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bondal A.: Symplectic groupoids related to Poisson–Lie groups. Tr. Mat. Inst. Steklova 246, 43–63 (2004)MathSciNetGoogle Scholar
  3. 3.
    Cattaneo, A.S., Felder, G.: Poisson sigma models and symplectic groupoids. In: Quantization of singular symplectic quotients, Progr. Math. 198, Basel: Birkhäuser, 2001, pp. 61–93Google Scholar
  4. 4.
    Chekhov L.O., Fock V.V.: Observables in 3d gravity and geodesic algebras. Czech. J. Phys. 50, 1201–1208 (2000)MathSciNetADSzbMATHCrossRefGoogle Scholar
  5. 5.
    Chekhov L., Mazzocco M.: Isomonodromic deformations and twisted Yangians arising in Teichmüller theory. Adv. Math. 226(6), 4731–4775 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Chekhov, L., Mazzocco, M.: Work in progress 2013Google Scholar
  7. 7.
    Crainic M., Fernandes R.: Integrability of Lie brackets. Ann. of Math. 157(2), 575–620 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Crainic M., Fernandes R.: Integrability of Poisson brackets. J. Diff. Geom. 66, 71–137 (2004)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Dubrovin, B.: Geometry of 2D topological field theories. In: Integrable systems and quantum groups (Montecatini Terme, 1993), Lecture Notes in Math. 1620, Berlin: Springer, 1996, pp. 120–348Google Scholar
  10. 10.
    Fernandes R., Iglesias D.: Integrability of Poisson–Lie Group Actions. Lett. Math. Phys. 90, 137–159 (2009)MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. 11.
    Fock V., Goncharov A.: Moduli spaces of local systems and higher Teichml̈ler theory. Publ. Math. IHES 103(1), 1–211 (2006)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Fock V., Marshakov A.: A note on quantum groups and relativistic Toda theory. Nucl. Phys. B. 56, 208–214 (1997)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Fock, V.V., Rosly, A.A.: Moduli space of flat connections as a Poisson manifold. In: Advances in quantum field theory and statistical mechanics: 2nd Italian-Russian collaboration (Como, 1996), Int. J. Mod. Phys. B 11, no. 26–27, 3195–3206 (1997)Google Scholar
  14. 14.
    Hitchin, N.: Deformations of holomorphic Poisson manifolds. [math.DG], 2011
  15. 15.
    Mackenzie, K.: General Theory of Lie Groupoids and Lie Algebroids. LMS Lect. Note Series 213, Cambridge: Cambridge Univ. Press, 2005Google Scholar
  16. 16.
    Molev, A.: Yangians and classical Lie algebras. Mathematical Surveys and Monographs. 143, Providence, RI: Amer. Math. Soc., 2007Google Scholar
  17. 17.
    Molev A., Ragoucy E.: Symmetries and invariants of twisted quantum algebras and associated Poisson algebras. Rev. Math. Phys. 20(2), 173–198 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Molev A., Ragoucy E., Sorba P.: Coideal subalgebras in quantum affine algebras. Rev. Math. Phys. 15, 789–822 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Nelson J.E., Regge T.: Homotopy groups and (2 + 1)-dimensional quantum gravity. Nucl. Phys. B 328, 190–199 (1989)MathSciNetADSCrossRefGoogle Scholar
  20. 20.
    Nelson J.E., Regge T., Zertuche F.: Homotopy groups and (2 + 1)-dimensional quantum de Sitter gravity. Nucl. Phys. B 339, 516–532 (1990)MathSciNetADSCrossRefGoogle Scholar
  21. 21.
    Noumi M.: Macdonald’s symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces. Adv. Math. 123(1), 16–77 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Ugaglia M.: On a Poisson structure on the space of Stokes matrices. Int. Math. Res. Not. 1999(9), 473–493 (1999)MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Steklov Mathematical Institute and Laboratoire PonceletMoscowRussia
  2. 2.School of MathematicsLoughborough UniversityLoughboroughUK

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