Poisson geometry of SL(3,ℂ)-character varieties relative to a surface with boundary

Abstract

The S L ( 3 , C ) \mathrm {SL}(3,\mathbb {C}) -representation variety R \mathfrak {R} of a free group F r \mathtt {F}_r arises naturally by considering surface group representations for a surface with boundary. There is an S L ( 3 , C ) \mathrm {SL}(3,\mathbb {C}) -action on the coordinate ring of R \mathfrak {R} by conjugation. The geometric points of the subring of invariants of this action is an affine variety X \mathfrak {X} . The points of X \mathfrak {X} parametrize isomorphism classes of completely reducible representations. We show the coordinate ring C [ X ] \mathbb {C}[\mathfrak {X}] is a complex Poisson algebra with respect to a presentation of F r \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.