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 SIGMA, 2007, Volume 3, 066 (Mi sigma192)

Teichmüller Theory of Bordered Surfaces

Leonid O. Chekhovabcd

a Institute for Theoretical and Experimental Physics, Moscow, Russia
b Steklov Mathematical Institute, Moscow, Russia
c Poncelet Laboratoire International Franco-Russe, Moscow, Russia
d Concordia University, Montréal, Quebec, Canada

Abstract: We propose the graph description of Teichmüller theory of surfaces with marked points on boundary components (bordered surfaces). Introducing new parameters, we formulate this theory in terms of hyperbolic geometry. We can then describe both classical and quantum theories having the proper number of Thurston variables (foliation-shear coordinates), mapping-class group invariance (both classical and quantum), Poisson and quantum algebra of geodesic functions, and classical and quantum braid-group relations. These new algebras can be defined on the double of the corresponding graph related (in a novel way) to a double of the Riemann surface (which is a Riemann surface with holes, not a smooth Riemann surface). We enlarge the mapping class group allowing transformations relating different Teichmüller spaces of bordered surfaces of the same genus, same number of boundary components, and same total number of marked points but with arbitrary distributions of marked points among the boundary components. We describe the classical and quantum algebras and braid group relations for particular sets of geodesic functions corresponding to $A_n$ and $D_n$ algebras and discuss briefly the relation to the Thurston theory.

Keywords: graph description of Teichmüller spaces; hyperbolic geometry; algebra of geodesic functions

DOI: https://doi.org/10.3842/SIGMA.2007.066

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Bibliographic databases:

ArXiv: math.AG/0610872
Document Type: Article
MSC: 37D40; 53C22
Received: January 5, 2007; in final form April 28, 2007; Published online May 15, 2007
Language: English

Citation: Leonid O. Chekhov, “Teichmüller Theory of Bordered Surfaces”, SIGMA, 3 (2007), 066, 37 pp.

Citation in format AMSBIB
\Bibitem{Che07}
\by Leonid O.~Chekhov
\paper Teichm\"uller Theory of Bordered Surfaces
\jour SIGMA
\yr 2007
\vol 3
\papernumber 066
\totalpages 37
\mathnet{http://mi.mathnet.ru/sigma192}
\crossref{https://doi.org/10.3842/SIGMA.2007.066}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2322793}
\zmath{https://zbmath.org/?q=an:1155.30359}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84889234698}

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. L. O. Chekhov, “Riemann Surfaces with Orbifold Points”, Proc. Steklov Inst. Math., 266 (2009), 228–250
2. Chekhov L.O., “Orbifold Riemann surfaces and geodesic algebras”, J. Phys. A, 42:30 (2009), 304007, 32 pp.
3. M. Mazzocco, L. O. Chekhov, “Orbifold Riemann surfaces: Teichmüller spaces and algebras of geodesic functions”, Russian Math. Surveys, 64:6 (2009), 1079–1130
4. Chekhov L., Mazzocco M., “Shear coordinate description of the quantized versal unfolding of a D-4 singularity”, Journal of Physics A-Mathematical and Theoretical, 43:44 (2010), 442002
5. Chekhov L., Mazzocco M., “Isomonodromic deformations and twisted Yangians arising in Teichmüller theory”, Adv Math, 226:6 (2011), 4731–4775
6. Kolb S., Pellegrini J., “Braid group actions on coideal subalgebras of quantized enveloping algebras”, J Algebra, 336:1 (2011), 395–416
7. Chekhov L., Mazzocco M., “Teichmüller Spaces as Degenerated Symplectic Leaves in Dubrovin-Ugaglia Poisson Manifolds”, Physica D, 241:23-24 (2012), 2109–2121
8. Xu Y., Yang Sh., “Pbw-Deformations of Quantum Groups”, J. Algebra, 408 (2014), 222–249
9. Chekhov L., Shapiro M., “Teichmüller Spaces of Riemann Surfaces With Orbifold Points of Arbitrary Order and Cluster Variables”, Int. Math. Res. Notices, 2014, no. 10, 2746–2772
10. Xu Y., Wang D., Chen J., “Analogues of Quantum Schubert Cell Algebras in Pbw-Deformations of Quantum Groups”, J. Algebra. Appl., 15:10 (2016), 1650179
11. Stancu A., “A Note on Commutative Weakly Nil Clean Rings”, J. Algebra. Appl., 15:10 (2016), 1620001
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