Cluster algebras and Weil-Petersson forms

Duke Mathematical Journal

Volumn 127, Issue 2, 2005, Pages 291-311

  Gekhtman, Michael ^a   Shapiro, Michael ^b   Vainshtein, Alek ^c  

^a UNIVERSITY OF NOTRE DAME   (United States)

^b MICHIGAN STATE UNIVERSITY   (United States)

^c UNIVERSITY OF HAIFA   (Israel)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 18144396883     PISSN: 00127094     EISSN: None     Source Type: Journal    
DOI: 10.1215/S0012-7094-04-12723-X     Document Type: Article

References (25)

1. [BZ] A. BERENSTEIN and A. ZELEVINSKY, Tensor product multiplicities, canonical bases and totally positive varieties, Invent. Math. 143 (2001), 77-128. MR 1802793
2. [B] YU. BURMAN, Triangulations of surfaces with boundary and the homotopy principle for functions without critical points, Ann. Global Anal. Geom. 17 (1999), 221-238. MR 1698458
3. [F] V. FOCK, Dual Teichmüller spaces, preprint, arXiv:dg-ga/9702018
4. [FG1] V. V. FOCK AND A. GONCHAROV, Cluster ensembles, quantization and the dilogarithm, preprint, arXiv:math.AG/0311245
5. [FG2] _, Moduli spaces of local systems and higher Teichmüller theory, preprint, arXiv:math.AG/0311149
6. [FR] 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. MR 1730456
7. [FZ1] S. FOMIN and A. ZELEVINSKY, Double Bruhat cells and total positivity, J. Amer. Math. Soc. 12 (1999), 335-380. MR 1652878
8. [FZ2] _, Cluster algebras, I: Foundations, J. Amer. Math. Soc. 15 (2002), 497-529. MR 1887642
9. [FZ3] _, The Laurent phenomenon, Adv. in Appl. Math. 28 (2002), 119-144. MR 1888840
10. [FZ4] _, Cluster algebras, II: Finite type classification, Invent. Math. 154 (2003), 63-121. MR 2004457
11. [GSV] M. GEKHTMAN, M. SHAPIRO, and A. VAINSHTEIN, Cluster algebras and Poisson geometry, Moscow Math. J. 3 (2003), 899-934. MR 2078567
12. [H] A. HATCHER, On triangulation of surfaces. Topology Appl. 40 (1991), 189-194. MR 1123262
13. [I] N. V. IVANOV, "Mapping class groups" in Handbook of Geometric Topology, North-Holland, Amsterdam, 2002, 523-633. MR 1886678
14. [K] R. M. KASHAEV, Quantization of Teichmller spaces and the quantum dilogarithm, Lett. Math. Phys. 43 (1998), 105-115. MR 1607296
15. [KZ] 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. MR 1916837
16. [P1] R. C. PENNER, The decorated Teichmüller space of punctured surfaces, Comm. Math. Phys. 113 (1987), 299-339. MR 0919235
17. [P2] _, Weil-Petersson volumes, J. Differential Geom. 35 (1992), 559-608. MR 1163449
18. [S] J. SCOTT, Grassmannians and cluster algebras, preprint, arXiv:math.CO/0311148
19. n (ℝ)/B, Internat. Math. Res. Notices 1997, no. 10, 469-493. MR 1446839
20. [SSV2] _, Skew-symmetric vanishing lattices and intersections of Schubert cells, Internat. Math. Res. Notices 1998, no. 11, 563-588. MR 1631773
21. [SSVZ] 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. MR 1786504
22. [T] W. THURSTON, Minimal stretch maps between hyperbolic surfaces, preprint, arXiv:math.GT/9801039
23. [W] D. WEST, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, N.J., 1996. MR 1367739
24. [Z1] A. ZELEVINSKY, Connected components of real double Bruhat cells, Internat. Math. Res. Notices 2000, no. 21, 1131-1154. MR 1800992
25. [Z2] _, "From Littlewood-Richardson coefficients to cluster algebras in three lectures" in Symmetric Functions 2001: Surveys of Developments and Perspectives, ed. S. Fomin, NATO Sci. Ser II Math. Phys. Chem. 74, Kluwer, Dordrecht, 2002, 253-273. MR 2059365