Poisson Geometry of SL(3,C)-Character Varieties Relative to a Surface with Boundary
Abstract
Description
The SL(3,C)-representation variety R of a free group F arises naturally by considering surface group representations for a surface with boundary. There is a SL(3,C)-action on the coordinate ring of R by conjugation. The geometric points of the subring of invariants of this action is an affine variety X. The points of X parametrize isomorphism classes of completely reducible representations. We show the coordinate ring C[X] is a complex Poisson algebra with respect to a presentation of F 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.
33 pages, 11 figures, many revisions and some corrections, this version to appear in Transactions of the AMS
33 pages, 11 figures, many revisions and some corrections, this version to appear in Transactions of the AMS