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

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.

arXiv comment33 pages, 11 figures, many revisions and some corrections, this version to appear in Transactions of the AMS

References in corpus (2)

• Generators, Relations and Symmetries in Pairs of 3x3 Unimodular Matrices
• Defining relations of invariants of two 3 \times 3 matrices

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• The Topology of Parabolic Character Varieties of Free Groups
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• Batalin-Vilkovisky structures on moduli spaces of flat connections
• Compact connected components in relative character varieties of punctured spheres
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• Non-ergodicity on SU(2) and SU(3) character varieties of the once-punctured torus
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