Computations of the Structure of the Goldman Lie Algebra for the Torus

Felicia Tabing

Abstract.

We consider the structure of the Goldman Lie algebra for the closed torus, and show that it is finitely generated over the rationals. We also consider other traditional Lie algebra structures and determine that the Goldman Lie algebra for the torus is not nilpotent or solvable, and we compute the derived Lie algebra.

1. The Goldman Lie Algebra

The Goldman Lie algebra is an algebra over the module generated by the set of free homotopy classes of loops on a surface, described by intersection and concatenation, that was introduced by William M. Goldman in 1986 [Go].
Throughout this paper, let $\Sigma_{g,n}$ denote an oriented, genus $g$ surface with $n\geq 0$ boundary components. Denote $\hat{\pi}(\Sigma_{g,n})$ to be the set of free homotopy classes of loops on $\Sigma_{g,n}$ , where we use $\hat{\pi}$ when it is clear from the context which fixed surface we are discussing. Recall the following.

Lemma 1.1.

The set of free homotopy classes of loops on a surface $\Sigma_{g,n}$ is in one-to-one correspondence with conjugacy classes of $\pi_{1}(\Sigma_{g,n})$ .