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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Poisson geometry of $ \mathrm{SL}(3,\mathbb{C})$-character varieties relative to a surface with boundary


Author: Sean Lawton
Journal: Trans. Amer. Math. Soc. 361 (2009), 2397-2429
MSC (2000): Primary 58D29; Secondary 14D20
Published electronically: December 16, 2008
MathSciNet review: 2471924
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Abstract: The $ \mathrm{SL}(3,\mathbb{C})$-representation variety $ \mathfrak{R}$ of a free group $ \mathtt{F}_r$ arises naturally by considering surface group representations for a surface with boundary. There is an $ \mathrm{SL}(3,\mathbb{C})$-action on the coordinate ring of $ \mathfrak{R}$ by conjugation. The geometric points of the subring of invariants of this action is an affine variety $ \mathfrak{X}$. The points of $ \mathfrak{X}$ parametrize isomorphism classes of completely reducible representations. We show the coordinate ring $ \mathbb{C}[\mathfrak{X}]$ is a complex Poisson algebra with respect to a presentation of $ \mathtt{F}_r$ 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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Additional Information

Sean Lawton
Affiliation: Department of Mathematics, Instituto Superior Técnico, Lisbon, Portugal
Address at time of publication: Department of Mathematics, University of Maryland, College Park, Maryland 20742
Email: slawton@math.ist.utl.pt

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04777-6
PII: S 0002-9947(08)04777-6
Received by editor(s): March 23, 2007
Published electronically: December 16, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.