Teichm\"uller 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 ${\mathrm{\Sigma }}_{g,s,0}$ of genus $g$ with $s=1,2$ holes and on a Riemann sphere ${\mathrm{\Sigma }}_{0,1,n}$ with one hole and $n$ orbifold points of order two. We show that the corresponding Teichm"uller spaces $\mathcal T_{g,s,0}$and$\mathcal T_{0,1,n}$ 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.