Abstract
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 m = N, 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.
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Bondal A.: A symplectic groupoid of triangular bilinear forms and the braid groups. Izv. Math. 68, 659–708 (2004)
Bondal A.: Symplectic groupoids related to Poisson–Lie groups. Tr. Mat. Inst. Steklova 246, 43–63 (2004)
Cattaneo, A.S., Felder, G.: Poisson sigma models and symplectic groupoids. In: Quantization of singular symplectic quotients, Progr. Math. 198, Basel: Birkhäuser, 2001, pp. 61–93
Chekhov L.O., Fock V.V.: Observables in 3d gravity and geodesic algebras. Czech. J. Phys. 50, 1201–1208 (2000)
Chekhov L., Mazzocco M.: Isomonodromic deformations and twisted Yangians arising in Teichmüller theory. Adv. Math. 226(6), 4731–4775 (2011)
Chekhov, L., Mazzocco, M.: Work in progress 2013
Crainic M., Fernandes R.: Integrability of Lie brackets. Ann. of Math. 157(2), 575–620 (2003)
Crainic M., Fernandes R.: Integrability of Poisson brackets. J. Diff. Geom. 66, 71–137 (2004)
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. 120–348
Fernandes R., Iglesias D.: Integrability of Poisson–Lie Group Actions. Lett. Math. Phys. 90, 137–159 (2009)
Fock V., Goncharov A.: Moduli spaces of local systems and higher Teichml̈ler theory. Publ. Math. IHES 103(1), 1–211 (2006)
Fock V., Marshakov A.: A note on quantum groups and relativistic Toda theory. Nucl. Phys. B. 56, 208–214 (1997)
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)
Hitchin, N.: Deformations of holomorphic Poisson manifolds. http://arxiv.org/abs/1105.4775v1 [math.DG], 2011
Mackenzie, K.: General Theory of Lie Groupoids and Lie Algebroids. LMS Lect. Note Series 213, Cambridge: Cambridge Univ. Press, 2005
Molev, A.: Yangians and classical Lie algebras. Mathematical Surveys and Monographs. 143, Providence, RI: Amer. Math. Soc., 2007
Molev A., Ragoucy E.: Symmetries and invariants of twisted quantum algebras and associated Poisson algebras. Rev. Math. Phys. 20(2), 173–198 (2008)
Molev A., Ragoucy E., Sorba P.: Coideal subalgebras in quantum affine algebras. Rev. Math. Phys. 15, 789–822 (2003)
Nelson J.E., Regge T.: Homotopy groups and (2 + 1)-dimensional quantum gravity. Nucl. Phys. B 328, 190–199 (1989)
Nelson J.E., Regge T., Zertuche F.: Homotopy groups and (2 + 1)-dimensional quantum de Sitter gravity. Nucl. Phys. B 339, 516–532 (1990)
Noumi M.: Macdonald’s symmetric polynomials as zonal spherical functions on some quantum homogeneous spaces. Adv. Math. 123(1), 16–77 (1996)
Ugaglia M.: On a Poisson structure on the space of Stokes matrices. Int. Math. Res. Not. 1999(9), 473–493 (1999)
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Communicated by Y. Kawahigashi
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Chekhov, L., Mazzocco, M. Poisson Algebras of Block-Upper-Triangular Bilinear Forms and Braid Group Action. Commun. Math. Phys. 322, 49–71 (2013). https://doi.org/10.1007/s00220-013-1757-3
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DOI: https://doi.org/10.1007/s00220-013-1757-3