Poisson geometry of $\mathrm{SL}(3,\mathbb{C})$-character varieties relative to a surface with boundary

The $\mathrm{SL}(3,\mathbb{C})$-representation variety $\mathfrak{R}$ of a free group $\mathtt{F}_r$ arises naturally by considering surface group representations for a surface with boundary. There is an $\mathrm{SL}(3,\mathbb{C})$-action on the coordinate ring of $\mathfrak{R}$ by conjugation. The geometric points of the subring of invariants of this action is an affine variety $\mathfrak{X}$. The points of $\mathfrak{X}$ parametrize isomorphism classes of completely reducible representations. We show the coordinate ring $\mathbb{C}[\mathfrak{X}]$ is a complex Poisson algebra with respect to a presentation of $\mathtt{F}_r$ imposed by the surface. Lastly, we work out the bracket on all generators when the surface is a three-holed sphere or a one-holed torus.