Teichmüller spaces as degenerated symplectic leaves in Dubrovin–Ugaglia Poisson manifolds

In this paper, we study the Goldman bracket between geodesic length functions both on a Riemann surface Σ g , s , 0 of genus g with s = 1 , 2 holes and on a Riemann sphere Σ 0 , 1 , n with one hole and n orbifold points of order two. We show that the corresponding Teichmüller spaces T g , s , 0 and T 0 , 1 , n are realised as real slices of degenerated symplectic leaves in the Dubrovin–Ugaglia Poisson algebra of upper-triangular matrices S with 1 on the diagonal.