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An algebra-level version of a link-polynomial identity of Lickorish

Published online by Cambridge University Press:  01 May 2008

MICHAEL J. LARSEN  and
ERIC C. ROWELL
MICHAEL J. LARSEN
Affiliation:
Department of Mathematics, Indiana University, Bloomington, IN 47405, U.S.A. e-mail: larsen@math.indiana.edu
ERIC C. ROWELL
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843, U.S.A. e-mail: rowell@math.tamu.edu
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Abstract

We establish isomorphisms between certain specializations of BMW algebras and the symmetric squares of Temperley–Lieb algebras. These isomorphisms imply a link-polynomial identity due to W. B. R. Lickorish. As an application, we compute the closed images of the irreducible braid group representations factoring over these specialized BMW algebras.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 144 , Issue 3 , May 2008 , pp. 623 - 638
Copyright
Copyright © Cambridge Philosophical Society 2008

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References

REFERENCES

[BWj]Birman, J. and Wajnryb, B.. Markov classes in certain finite quotients of Artin's braid group. Israel J. Math. 56 (1986), no. 2, 160–178.CrossRefGoogle Scholar
[BWz]Birman, J. and Wenzl, H.. Braids, link polynomials and a new algebra. Trans. Amer. Math. Soc. 313 (1989), no. 1, 249–273.CrossRefGoogle Scholar
[FLW]Freedman, M. H., Larsen, M. J. and Wang, Z.. The two-eigenvalue problem and density of Jones representation of braid groups. Comm. Math. Phys. 228 (2002), 177–199, arXiv: math.GT/0103200.CrossRefGoogle Scholar
[GW]Goodman, F. and Wenzl, H.. The Temperley–Lieb algebra at roots of unity. Pacific J. Math. 161 (1993), no. 2, 307–334.CrossRefGoogle Scholar
[Gt]Goursat, E.. Sur les substitutions orthogonales et les divisions réguliéres de l'espace. Ann. Sci. École Norm. Sup. (3) 6 (1889), 9–102.CrossRefGoogle Scholar
[J1]F, V.. Jones, R.. Braid groups, Hecke algebras and type II sb 1 factors, Geometric Methods in Operator Algebras (Kyoto, 1983), 242–273, Pitman Res. Notes Math. Ser. 123 (Longman Sci. Tech., 1986).Google Scholar
[J2]F, V.. Jones, R.. On a certain value of the Kauffman polynomial. Comm. Math. Phys. 125 (1989), no. 3, 459–467.Google Scholar
[J3]F, V.. Jones, R.. Subfactors and Knots. CBMS Regional Conference Series in Mathematics, 80. (Amer. Math. Soc. 1991).CrossRefGoogle Scholar
[K]Kauffman, L.. An invariant of regular isotopy. Trans. Amer. Math. Soc. 318 (1990), no. 2, 417–471.CrossRefGoogle Scholar
[LRW]J, M.. Larsen, Rowell, E. C. and Wang, Z.. The N-eigenvalue problem and two applications. Int. Math. Res. Not. 2005 (2005), no. 64, 3987–4018.Google Scholar
[Li]B, W.. Lickorish, R.. Some link-polynomial relations. Math. Proc. Camb. Phil. Soc. 105 (1989), no. 1, 103–107.Google Scholar
[M]Murakami, J.. The Kauffman polynomial of links and representation theory. Osaka J. Math. 24 (1987), no. 4, 745–758.Google Scholar
[R1]Rowell, E. C.. On a family of non-unitarizable ribbon categories: Math Z. 250 no. 4 (2005), 745–774.CrossRefGoogle Scholar
[S]Sundaram, S.. On the combinatorics of representations of Sp(2n, ). Ph. D. Thesis (M.I.T. 1986).Google Scholar
[Wz1]Wenzl, H.. Hecke algebras of type An and subfactors. Invent. Math. 92 (1988), 349–383.CrossRefGoogle Scholar
[Wz2]Wenzl, H.. Quantum groups and subfactors of type B, C and D. Comm. Math. Phys. 133 (1990), no. 2, 383–432.CrossRefGoogle Scholar