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Let $\Sigma_{g}$ be a closed oriented surface of genus $g$, Goldman defined a Lie algebra structure on the free module generated by the free homotopy classes of loops on $\Sigma_{g}$. Roughly speaking, the Lie bracket is defined via intersection and concatenation of loops. In detail, see his paper Invariant functions on Lie groups and Hamiltonian flows of surface group representations .

I want to know that are there any formulas to compute this Lie bracket for small nonzero $g$? For example, what is the Goldman Lie algebra of torus? Any help or any references would be very appreciated.

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I would recommend looking at the work of Moira Chas to start.

Here are two interesting papers of hers to read:

The Goldman bracket and the intersection of curves on surfaces.

Combinatorial Lie bialgebras of curves on surfaces

She even has an app on her website that computes the bracket for you: Goldman Bracket.

How to use the app: in the first box you input the relation defining the surface group (so for a torus abAB or a pair-of-pants abc). The next two boxes are for words in the surface group representing (free) homotopy classes of closed curves. For the syntax, lower case letters like a,b,c are for generators of the surface group and capital letters like A,B,C are for the inverses of a,b,c, respectively. I believe the app expects cyclically reduced words.

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  • $\begingroup$ Thanks a lot. By the way, do you know how to use her app? I am quite confused about the inputs. $\endgroup$
    – Yining Zhang
    Commented Jun 7, 2018 at 2:45
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    $\begingroup$ @YiningZhang Why don't you ask her? $\endgroup$
    – Igor Rivin
    Commented Jun 7, 2018 at 4:11

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