Isomonodromic deformations and twisted Yangians arising in Teichmuller theory

In this paper we build a link between the Teichmuller 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 Poincare uniformization. In the case of a one-sheeted hyperboloid with n orbifold points we show that the Poisson algebra D-n of geodesic length functions is the semiclassical limit of the twisted q-Yangian Y-q(t) (o(n)) for the orthogonal Lie algebra o(n) defined by Molev, Ragoucy and Sorba. We give a representation of the braid-group action on D-n 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 D-n as the Poisson algebra of monodromy data of a Frobenius manifold in the vicinity of a non-semi-simple point. (C) 2010 Elsevier Inc. All rights reserved.