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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



$SL_n$-character varieties as spaces of graphs

Author: Adam S. Sikora
Journal: Trans. Amer. Math. Soc. 353 (2001), 2773-2804
MSC (1991): Primary 20C15, 57M27
Published electronically: March 15, 2001
MathSciNet review: 1828473
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Abstract | References | Similar Articles | Additional Information


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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  • [BHMV-1] C. Blanchet, N. Habegger, G. Masbaum, P. Vogel, Three-manifold invariants derived from the Kauffman bracket, Topology, 31 (1992), no. 4, 685-699. MR 94a:57010
  • [BHMV-2] C. Blanchet, N. Habegger, G. Masbaum, P. Vogel, Topological quantum field theories derived from the Kauffman bracket, Topology, 34 (1995), no. 4, 883-927. MR 96i:57015
  • [B-H] G. W. Brumfiel, H. M. Hilden, $SL(2)$ Representations of Finitely Presented Groups, Contemp. Math. 187 (1995). MR 96g:20004
  • [B-1] D. Bullock, Estimating a skein module with $SL_2({\mathbb C})$ characters, Proc. Amer. Math. Soc. 125(6), 1997, 1835-1839. MR 97g:57018
  • [B-2] D. Bullock, Rings of $SL_2({\mathbb C})$-characters and the Kauffman bracket skein module, Comment. Math. Helv. 72, 1997, 521-542. MR 98k:57008
  • [Co] J. H. Conway, An enumeration of knots and links, and some of their algebraic properties. Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967), pp. 329-358, Pergamon, Oxford, 1970. MR 41:2661
  • [Cv] P. Cvitanovic, Group Theory, Part I, Nordita Notes, January 1984.
  • [Fog] J. Fogarty, Invariant Theory, W. A. Benjamin, Inc. 1969. MR 29:1458
  • [For] E. Formanek, The Polynomial Identities and Invariants of $n\times n$ Matrices, CBMS Regional Conf. Ser. 78 AMS, 1991. MR 92g:16031
  • [FYHLMO] P. Freyd, D. Yetter, J. Hoste, W.B.R. Lickorish, K. Millett, A. Ocneanu, A new polynomial invariant of knots and links. Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 2, 239-246. MR 86e:57007
  • [H-P] J. Hoste, J.H. Przytycki, A survey of skein modules of 3-manifolds; in Knots 90, Proceedings of the International Conference on Knot Theory and Related Topics, Osaka (Japan), August 15-19, 1990, Editor A. Kawauchi, Walter de Gruyter (1992), 363-379. MR 93m:57018
  • [Jo] V. F. R. Jones, Hecke algebra representations of braid groups and link polynomials. Ann. of Math. 126 (1987), no. 2, 335-388. MR 89c:46092
  • [KM] M. Kapovich, J. J. Millson, On representation varieties of Artin groups, projective arrangements and the fundamental groups of smooth complex algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 88, (1998), 5-95 (1999). CMP 2000:07
  • [Ka] L. H. Kauffman, An invariant of regular isotopy, Trans. Amer. Math. Soc. 318 (1990), no. 2, 417-471. MR 90g:57007
  • [L1] W. B. R. Lickorish, Three-manifolds and the Temperley-Lieb algebra, Math. Ann. 290 (1991), no. 4, 657-670. MR 92e:57014
  • [L2] W. B. R. Lickorish, Calculations with the Temperley-Lieb algebra, Comment. Math. Helv. 67 (1992), no. 4, 571-591. MR 94c:57032
  • [L3] W. B. R. Lickorish, The skein method for three-manifold invariants, J. Knot Theory Ramifications 2 (1993), no. 2, 171-194. MR 94g:57006
  • [L-M] A. Lubotzky, A. Magid, Varieties of representations of finitely generated groups, Memoirs of the AMS 336 (1985). MR 87c:20021
  • [MFK] D. Mumford, J. Fogarty, F. Kirwan Geometric Invariant Theory, Springer-Verlag 1994. MR 95m:14012
  • [Ny] L. Nyssen, Pseudo-représentations, Mathematische Annalen 306 (1996), 257-283. MR 98a:20013
  • [Pro-1] C. Procesi, The invariant theory of $n\times n$matrices, Adv. in Math. 19 (1976), 301-381. MR 54:7512
  • [Pro-2] C. Procesi, A formal inverse to the Cayley-Hamilton theorem, J. of Alg. 107 (1987), 63-74. MR 88b:16033
  • [PS-1] J.H. Przytycki, A.S. Sikora, Skein algebra of a group, Proceedings of an International Conference in Knot Theory, Warsaw 1995, Banach Center Publications, 42, Polish Acad. Sci., Warsaw, 1998, pp. 297-306. MR 99e:57019
  • [PS-2] J.H. Przytycki, A.S. Sikora, On Skein Algebras and $Sl_2({\mathbb C})$-character varieties, Topology. 39 (2000), 115-148. MR 2000g:57026
  • [PS-3] J.H. Przytycki, A.S. Sikora, Skein algebras of surfaces, Preprint.
  • [P-T] J. H. Przytycki, P. Traczyk, Invariants of links of Conway type, Kobe J. Math. 4 (1988), no. 2, 115-139. MR 89h:57006
  • [Ra] Y. P. Razmyslov, Identities of algebras and their representations, Translations of Mathematical Monographs, 138, AMS 1994. MR 95i:16022
  • [R-T] N. Reshetikhin, V. G. Turaev, Invariants of $3$-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), no. 3, 547-597. MR 92b:57024
  • [Ro-1] J. Roberts, Skein theory and Turaev-Viro invariants, Topology 34 (1995), no. 4, 771-787. MR 97g:57014
  • [Ro-2] J. Roberts, Skeins and mapping class groups, Math. Proc. Cambridge Philos. Soc. 115 (1994), no. 1, 53-77. MR 94m:57035
  • [Ro] R. Rouquier, Caractérisation des caractèrs et Pseudo-caractères, J. of Algebra 180, no. 0083, (1996) 571-586. MR 97a:20010
  • [Sa-1] K. Saito, Representation varieties of a finitely generated group into $SL_2$ or $GL_2,$ RIMS Publications 958, Kyoto University (1993).
  • [Sa-2] K. Saito, Character variety of representations of a finitely generated group in $SL_2,$ Topology and Teichmüller Spaces (Proc. Taniguchi Sympos., Katinkulta, Finland, 1995; S. Kojima et al., eds.), World Sci. Publ., Singapore, 1996, pp. 253-264. MR 99k:20011
  • [Si] A.S. Sikora, Spin networks and character varieties, in preparation.
  • [Ta] R. Taylor, Galois representations associated to Siegel modular forms of low weight, Duke Math. Journal 63 (1991), no. 2, 281-332. MR 92j:11044
  • [We] H. Weyl, The Classical Groups, Princeton Univ. Press, 1939. MR 1:42c
  • [Yo] Y. Yokota, Skeins and quantum ${SU}(N)$ invariants of $3$-manifolds. Math. Ann. 307 (1997), no. 1, 109-138. MR 98h:57039

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Additional Information

Adam S. Sikora
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742

Keywords: Character, character variety, skein module
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
Article copyright: © Copyright 2001 American Mathematical Society

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