Duke Mathematical Journal

Cluster algebras and Weil-Petersson forms

Michael Gekhtman, Michael Shapiro, and Alek Vainshtein

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In our paper [GSV], we discussed Poisson properties of cluster algebras of geometric type for the case of a nondegenerate matrix of transition exponents. In this paper, we consider the case of a general matrix of transition exponents. Our leading idea is that a relevant geometric object in this case is a certain closed 2-form compatible with the cluster algebra structure. The main example is provided by Penner coordinates on the decorated Teichmüller space, in which case the above form coincides with the classical Weil-Petersson symplectic form.

Article information

Source
Duke Math. J., Volume 127, Number 2 (2005), 291-311.

Dates
First available in Project Euclid: 23 March 2005

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1111609853

Digital Object Identifier
doi:10.1215/S0012-7094-04-12723-X

Mathematical Reviews number (MathSciNet)
MR2130414

Zentralblatt MATH identifier
1079.53124

Subjects
Primary: 53D17: Poisson manifolds; Poisson groupoids and algebroids
Secondary: 53D30: Symplectic structures of moduli spaces 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx] 14M20: Rational and unirational varieties [See also 14E08]

Citation

Gekhtman, Michael; Shapiro, Michael; Vainshtein, Alek. Cluster algebras and Weil-Petersson forms. Duke Math. J. 127 (2005), no. 2, 291--311. doi:10.1215/S0012-7094-04-12723-X. https://projecteuclid.org/euclid.dmj/1111609853


Export citation

References

  • [1] A. Berenstein and A. Zelevinsky, Tensor product multiplicities, canonical bases and totally positive varieties, Invent. Math. 143 (2001), 77--128.
  • [2] Yu. Burman, Triangulations of surfaces with boundary and the homotopy principle for functions without critical points, Ann. Global Anal. Geom. 17 (1999), 221--238.
  • [3] V. Fock, Dual Teichmüller spaces.
  • [4] V. V. Fock and A. Goncharov, Cluster ensembles, quantization and the dilogarithm.
  • [5] —, Moduli spaces of local systems and higher Teichmüller theory.
  • [6] V. Fock and A. Rosly, ``Poisson structure on moduli of flat connections on Riemann surfaces and the $r$-matrix'' in Moscow Seminar in Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2 191, Amer. Math. Soc., Providence, 1999, 67--86.
  • [7] S. Fomin and A. Zelevinsky, Double Bruhat cells and total positivity, J. Amer. Math. Soc. 12 (1999), 335--380.
  • [8] —, Cluster algebras, I: Foundations, J. Amer. Math. Soc. 15 (2002), 497--529.
  • [9] —, The Laurent phenomenon, Adv. in Appl. Math. 28 (2002), 119--144.
  • [10] —, Cluster algebras, II: Finite type classification, Invent. Math. 154 (2003), 63--121.
  • [11] M. Gekhtman, M. Shapiro, and A. Vainshtein, Cluster algebras and Poisson geometry, Moscow Math. J. 3 (2003), 899--934.
  • [12] A. Hatcher, On triangulation of surfaces, Topology Appl. 40 (1991), 189--194.
  • [13] N. V. Ivanov, ``Mapping class groups'' in Handbook of Geometric Topology, North-Holland, Amsterdam, 2002, 523--633.
  • [14] R. M. Kashaev, Quantization of Teichmüller spaces and the quantum dilogarithm, Lett. Math. Phys. 43 (1998), 105--115.
  • [15] 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.
  • [16] R. C. Penner, The decorated Teichmüller space of punctured surfaces, Comm. Math. Phys. 113 (1987), 299--339.
  • [17] —, Weil-Petersson volumes, J. Differential Geom. 35 (1992), 559--608.
  • [18] J. Scott, Grassmannians and cluster algebras.
  • [19] B. Shapiro, M. Shapiro, and A. Vainshtein, Connected components in the intersection of two open opposite Schubert cells in $\SL_n(\mathbbR)/B$, Internat. Math. Res. Notices 1997, no. 10, 469--493.
  • [20] —, Skew-symmetric vanishing lattices and intersections of Schubert cells, Internat. Math. Res. Notices 1998, no. 11, 563--588.
  • [21] B. Shapiro, M. Shapiro, A. Vainshtein, and A. Zelevinsky, Simply laced Coxeter groups and groups generated by symplectic transvections, Michigan Math. J. 48 (2000), 531--551.
  • [22] W. Thurston, Minimal stretch maps between hyperbolic surfaces.
  • [23] D. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, N.J., 1996.
  • [24] A. Zelevinsky, Connected components of real double Bruhat cells, Internat. Math. Res. Notices 2000, no. 21, 1131--1154.