Poisson Geometry of SL(3,C)-Character Varieties Relative to a Surface with Boundary
Symplectic Geometry
2009-03-16 v2 Algebraic Geometry
Abstract
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.
Keywords
Cite
@article{arxiv.math/0703251,
title = {Poisson Geometry of SL(3,C)-Character Varieties Relative to a Surface with Boundary},
author = {Sean Lawton},
journal= {arXiv preprint arXiv:math/0703251},
year = {2009}
}
Comments
33 pages, 11 figures, many revisions and some corrections, this version to appear in Transactions of the AMS