$SL_n$-character varieties as spaces of graphs
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- by Adam S. Sikora PDF
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Abstract:
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 Feynman diagrams) in any topological space $X,$ with $\pi _1(X)=G.$ We also show that all such relations are implied by a single local relation between graphs. In this way, we provide a topological approach to the study of $SL_n$-representations of groups.
The motivation for this paper was our work with J. Przytycki on invariants of links in 3-manifolds which are based on the Kauffman bracket skein relation. These invariants lead to a notion of a skein module of $M$ which, by a theorem of Bullock, Przytycki, and the author, is a deformation of the $SL_2$-character variety of $\pi _1(M).$ This paper provides a generalization of this result to all $SL_n$-character varieties.
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Additional Information
- Adam S. Sikora
- Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
- MR Author ID: 364939
- Email: asikora@math.umd.edu
- Received by editor(s): May 18, 1999
- Received by editor(s) in revised form: August 9, 2000
- Published electronically: March 15, 2001
- Additional Notes: Partially supported by NSF grant DMS93-22675
- © Copyright 2001 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 353 (2001), 2773-2804
- MSC (1991): Primary 20C15, 57M27
- DOI: https://doi.org/10.1090/S0002-9947-01-02700-3
- MathSciNet review: 1828473