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19
Symplectic manifolds and isomonodromic deformations
 ADV. MATH
"... We study moduli spaces of meromorphic connections (with arbitrary order poles) over Riemann surfaces together with the corresponding spaces of monodromy data (involving Stokes matrices). Natural symplectic structures are found and described both explicitly and from an infinite dimensional viewpoint ..."
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Cited by 43 (6 self)
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We study moduli spaces of meromorphic connections (with arbitrary order poles) over Riemann surfaces together with the corresponding spaces of monodromy data (involving Stokes matrices). Natural symplectic structures are found and described both explicitly and from an infinite dimensional viewpoint (generalising the AtiyahBott approach). This enables us to give an intrinsic symplectic description of the isomonodromic deformation equations of Jimbo, Miwa and Ueno, thereby putting the existing results for the six Painleve ́ equations and Schlesinger’s equations into a uniform framework.
Stokes matrices, Poisson Lie groups and Frobenius Manifolds
 INVENT. MATH. 146, 479–506
, 2001
"... The purpose of this paper is to point out and then draw some consequences of the fact that the Poisson Lie group G ∗ dual to G = GLn(C) may be identified with a certain moduli space of meromorphic connections over the unit disc having an irregular singularity at the origin. (G ∗ will be fully ..."
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Cited by 17 (5 self)
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The purpose of this paper is to point out and then draw some consequences of the fact that the Poisson Lie group G ∗ dual to G = GLn(C) may be identified with a certain moduli space of meromorphic connections over the unit disc having an irregular singularity at the origin. (G ∗ will be fully
A quantum duality principle for coisotropic subgroups and Poisson quotients,
 Advances in Mathematics
, 2006
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Teichmüller theory of bordered surfaces
, 2006
"... 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 ..."
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Cited by 8 (5 self)
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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 (foliationshear coordinates), mappingclass group invariance (both classical and quantum), Poisson and quantum algebra of geodesic functions, and classical and quantum braidgroup 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 An and Dn algebras and discuss briefly the relation to the Thurston theory. 1
Orbifold Riemann surfaces and geodesic algebras
 J. Phys. A: Math. Theor
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ISOMONODROMIC DEFORMATIONS AND TWISTED YANGIANS ARISING IN TEICHMÜLLER THEORY
, 2009
"... In this paper we build a link between the Teichmüller theory of hyperbolic Riemann surfaces and isomonodromic deformations of linear systems whose monodromy group is the Fuchsian group associated to the given hyperbolic Riemann surface by the Poincaré uniformization. In the case of a one–sheeted hyp ..."
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Cited by 6 (4 self)
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In this paper we build a link between the Teichmüller theory of hyperbolic Riemann surfaces and isomonodromic deformations of linear systems whose monodromy group is the Fuchsian group associated to the given hyperbolic Riemann surface by the Poincaré uniformization. In the case of a one–sheeted hyperboloid with n orbifold points we show that the Poisson algebra Dn of geodesic length functions is the semiclassical limit of the twisted q–Yangian Y ′ q (on) for the orthogonal Lie algebra on defined by Molev, Ragoucy and Sorba. We give a representation of the braid group action on Dn in terms of an adjoint matrix action. We characterize two types of finite–dimensional Poissonian reductions and give an explicit expression for the generating function of their central elements. Finally, we interpret the algebra Dn as the Poisson algebra of monodromy data of a Frobenius manifold in the vicinity of
Symmetries and invariants of twisted quantum algebras and associated Poisson algebras
, 2007
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STOKES MATRICES AND POISSON LIE GROUPS
, 2000
"... The purpose of this paper is to point out and then draw some consequences of the fact that the Poisson Lie group G ∗ dual to G = GLn(C) may be identified with a certain moduli space of meromorphic connections over the unit disc having an irregular singularity ..."
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Cited by 1 (0 self)
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The purpose of this paper is to point out and then draw some consequences of the fact that the Poisson Lie group G ∗ dual to G = GLn(C) may be identified with a certain moduli space of meromorphic connections over the unit disc having an irregular singularity