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

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

2007-03-09

arxiv.org/abs/math/0703251v2

Trans. Amer. Math. Soc. 361 (2009), 2397-2429

Mathematics

Symplectic Geometry

33 pages, 11 figures, many revisions and some corrections, this version to appear in Transactions of the AMS

Scientific paper

10.1090/S0002-9947-08-04777-6

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.

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