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

@article{Lawton2007PoissonGO,
title={Poisson geometry of "SL" (3, C)-character varieties relative to a surface with boundary},
author={Sean Lawton},
journal={Transactions of the American Mathematical Society},
year={2007},
volume={361},
pages={2397-2429}
}
• S. Lawton
• Published 9 March 2007
• Mathematics
• Transactions of the American Mathematical Society
The SL(3, C)-representation variety R of a free group F r arises naturally by considering surface group representations for a surface with boundary. There is an 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 r…

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## References

SHOWING 1-10 OF 67 REFERENCES
MAPPING CLASS GROUP DYNAMICS ON SURFACE GROUP REPRESENTATIONS
Deformation spaces Hom(�,G)/G of representations of the fundamental groupof a surfacein a Lie group G ad- mit natural actions of the mapping class group Mod�, preserving a Poisson structure. When G
Spin networks and SL(2,C)-Character varieties
• Mathematics
• 2005
Denote the free group on 2 letters by F_2 and the SL(2,C)-representation variety of F_2 by R=Hom(F_2,SL(2,C)). The group SL(2,C) acts on R by conjugation. We construct an isomorphism between the
The deformation theory of representations of fundamental groups of compact Kähler manifolds
• Mathematics
• 1988
AbstractLet Γ be the fundamental group of a compact Kähler manifold M and let G be a real algebraic Lie group. Let ℜ(Γ, G) denote the variety of representations Γ → G. Under various conditions on ρ ∈
Group systems, groupoids, and moduli spaces of parabolic bundles
• Mathematics
• 1995
Moduli spaces of homomorphisms or, more generally, twisted homomorphisms from fundamental groups of surfaces to compact connected Lie groups, were connected with geometry through their identification
GEOMETRY OF MODULI SPACES OF FLAT BUNDLES ON PUNCTURED SURFACES
For a Riemann surface with one puncture we consider moduli spaces of flat connections such that the monodromy transformation around the puncture belongs to a given conjugacy class with the property
$SL_n$-character varieties as spaces of graphs
An SL_n-character of a group G is the trace of an SL_n-representation of G. We show that all algebraic relations between SL_n-characters of G can be visualized as relations between graphs (resembling
Finite dimensional representations of algebras
Let R be a ring, K a field, and n a natural number. We will be concerned with the following type of questions: (a) Classify the representations of R in (hO, (or n dimensional representations). (b)
The ring of invariants of matrices
We denote by M(n) the space of all n × n-matrices with their coefficients in the complex number field C and by G the group of invertible matrices GL(n, C). Let W = M(n) i be the vector space of