# Lie bracket on the complex valued functions of the space of representations of a Riemann surface

Let $$S$$ be a closed surface and $$G$$ be a reductive Lie group. Goldman (Invariant functions on Lie groups and Hamiltonian flows of surface group representations) proved that, for a fairly general class of groups $$G$$, $$M=Hom(\pi_1(S),G)/G$$ admits a symplectic structure, where the quotient is by conjugation action. Hence the space of all "real valued" functions $$C^\infty(M,\mathbb R)$$ admits a Lie bracket $$\{\,,\}$$.

Suppose $$G$$ is one of the following Lie groups: $$\operatorname{GL}_n(\mathbb R)$$, $$\operatorname{GL}_n(\mathbb C)$$, $$\operatorname{SL}_n(\mathbb R)$$, or $$\operatorname{SL}_n(\mathbb C)$$. Given any $$x\in\pi_1(S)$$, we define a function $$f_x:M\to \mathbb R$$ by $$f_x(\rho)=\Re(\operatorname{tr}(\rho(x)))$$, where $$\Re$$ is the real-part of a complex number. Given $$x,y\in\pi_1(S)$$, Goldman gave explicit formulas for $$\{f_x, f_y\}$$.

In some papers, the authors consider the Lie bracket between two complex-valued function on $$M$$, and use Goldman's formula and paper as a reference. For example Section 4 of Andersen, Mattes, and Reshetikhin - The Poisson structure on the moduli space of flat connections and chord diagrams, Page 542 of Bishwas and Guruprasad - Principal bundles on open surfaces and invariant functions on Lie groups, and Etingof Casimirs of the Goldman Lie algebra of a closed surface, considered the Lie bracket of the trace functions (not just the real part) defined similarly as above.

My question is: what is the Lie bracket of $$C^\infty(M, \mathbb C)$$, and how is it related to the Lie bracket of $$C^\infty(M, \mathbb R)$$?

• In "Given $x, y \in \pi_1(S)$, Goldman gave explicit formulas for $\{f, g\}$", should that be "… for $\{f_x, f_y\}$"? May 5, 2019 at 18:23
In the complex case, Goldman's symplectic form is a $$(2,0)$$-form, and the Poisson bracket is in terms of that form (the bracket is determined by a bivector and the bivector corresponds to the form).