Casimirs of the Goldman Lie Algebra of a Closed Surface

Let Σ be a connected closed oriented surface of genus g. In 1986 Goldman [Go] attached to Σ a Lie algebra L = L(Σ), later shown by Turaev ([Tu]) to have a natural structure of a Lie bialgebra. It is defined as follows. As a vector space, L has a basis eγ labeled by conjugacy classes γ in the fundamental group π1(Σ), geometrically represented by closed oriented curves on Σ without a base point. To define the commutator [eγ1 , eγ2 ], one needs to bring the two curves γ1, γ2 into general position by isotopy, and then for each intersection point pi of the two curves, define γ3i to be the curve obtained by tracing γ1 and then γ2 starting and ending at pi. Then one defines [eγ1 , eγ2 ] to be ∑ i εieγ3i , where εi = 1 if γ1 approaches γ2 from the right at pi (with respect to the orientation of Σ), and −1 otherwise. The combinatorial structure of L has been much studied; see e.g. [C, Tu]. However, many problems about the structure of L remained open. In particular, in 2001, M. Chas and D. Sullivan communicated to me the following conjecture.