1
$\begingroup$

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 $C^\infty(M, \mathbb R)$?

Any kind of suggestion/reference/comment will be extremely helpful. Thanks in advance.

$\endgroup$
1
  • 1
    $\begingroup$ In "Given $x, y \in \pi_1(S)$, Goldman gave explicit formulas for $\{f, g\}$", should that be "… for $\{f_x, f_y\}$"? $\endgroup$ – LSpice May 5 '19 at 18:23
3
$\begingroup$

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).

The Goldman bracket formula works the same way as it does in the real case. You can find a proof about this latter fact, and concrete examples, in my paper:

Poisson Geometry of SL(3,C)-Character Varieties Relative to a Surface with Boundary.

As far as learning about general complex Poisson structures, relevant to this setting, I recommend:

Poisson Structures by Laurent-Gengoux, Pichereau, & Vanhaecke.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.