How are the Goldman Lie algebra of a closed surface $\overline{S}$ and the bordered surface $S$ obtained by taking $\overline{S}$ and removing an open disc (or more generally, $n$ disjoint discs) related?



Goldman Lie algebra of a bordered surface vs. a closed surface?
I am not really sure what you are asking, but the main theorem in Section 5.17 in Goldman's paper [Invariant functions on Lie groups and Hamiltonian flows of surface group representations](http://www.math.umd.edu/~wmg/InvariantFunctions.pdf) seems to imply that adding boundary components to $\overline{S}$ introduces central elements to the Lie algebra (but other changes occur too since the fundamental group becomes free).


However, quotienting by the ideal generated by the differences of these new central elements and the homotopy class of the identity (or any scalar really) will result in another Lie algebra which should be isomorphic to the original Lie algebra in terms of the closed surface.


This should be in parallel with the fact that the Lie algebra on free homotopy classes in $\overline{S}$ induces a symplectic structure on the (smooth locus of the) moduli space of homorphisms from $\pi\_1(\overline{S})$ to a Lie group $G$ (admitting an Ad-invariant bilinear form, for example reductive) and introducing boundaries to $\overline{S}$ turns the symplectic structure into Poisson structure. Fixing the invariant functions in terms of the boundary classes (the resulting Casimirs) to be the value at the identity then give a symplectic leaf isomorphic to the original symplectic moduli space of the closed surface.


