New subfactors from braid group representations

Author:
Juliana Erlijman

Journal:
Trans. Amer. Math. Soc. **350** (1998), 185-211

MSC (1991):
Primary 46L37

MathSciNet review:
1443192

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Abstract: This paper is about the construction of new examples of pairs of subfactors of the hyperfinite II factor, and the computation of their indices and relative commutants. The construction is done in general by considering unitary braid representations with certain properties that are satisfied in natural examples. We compute the indices explicitly for the particular cases in which the braid representations are obtained in connection with representation theory of Lie algebras of types A,B,C,D.

**[Bi]**Joan S. Birman,*Braids, links, and mapping class groups*, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974. Annals of Mathematics Studies, No. 82. MR**0375281****[BW]**Joan S. Birman and Hans Wenzl,*Braids, link polynomials and a new algebra*, Trans. Amer. Math. Soc.**313**(1989), no. 1, 249–273. MR**992598**, 10.1090/S0002-9947-1989-0992598-X**[Ch]**Marie Choda,*Index for factors generated by Jones’ two sided sequence of projections*, Pacific J. Math.**139**(1989), no. 1, 1–16. MR**1010781****[D]**V. G. Drinfel′d,*Quantum groups*, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 798–820. MR**934283****[FH]**William Fulton and Joe Harris,*Representation theory*, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR**1153249****[GHJ]**Frederick M. Goodman, Pierre de la Harpe, and Vaughan F. R. Jones,*Coxeter graphs and towers of algebras*, Mathematical Sciences Research Institute Publications, vol. 14, Springer-Verlag, New York, 1989. MR**999799****[GW]**Frederick M. Goodman and Hans Wenzl,*Littlewood-Richardson coefficients for Hecke algebras at roots of unity*, Adv. Math.**82**(1990), no. 2, 244–265. MR**1063959**, 10.1016/0001-8708(90)90090-A**[Hu]**James E. Humphreys,*Introduction to Lie algebras and representation theory*, Springer-Verlag, New York-Berlin, 1972. Graduate Texts in Mathematics, Vol. 9. MR**0323842****[J]**V. F. R. Jones,*Index for subfactors*, Invent. Math.**72**(1983), no. 1, 1–25. MR**696688**, 10.1007/BF01389127**[Ji]**Michio Jimbo,*Quantum 𝑅 matrix for the generalized Toda system*, Comm. Math. Phys.**102**(1986), no. 4, 537–547. MR**824090****[M]**Jun Murakami,*The Kauffman polynomial of links and representation theory*, Osaka J. Math.**24**(1987), no. 4, 745–758. MR**927059****[Mi]**Willard Miller Jr.,*Symmetry groups and their applications*, Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 50. MR**0338286****[O]**Adrian Ocneanu,*Chirality for operator algebras*, Subfactors (Kyuzeso, 1993) World Sci. Publ., River Edge, NJ, 1994, pp. 39–63. MR**1317353****[Po1]**Sorin Popa,*Orthogonal pairs of ∗-subalgebras in finite von Neumann algebras*, J. Operator Theory**9**(1983), no. 2, 253–268. MR**703810****[Po2]**S. Popa,*Classification of subfactors: the reduction to commuting squares*, Invent. Math.**101**(1990), no. 1, 19–43. MR**1055708**, 10.1007/BF01231494**[W-1]**Hans Wenzl,*Hecke algebras of type 𝐴_{𝑛} and subfactors*, Invent. Math.**92**(1988), no. 2, 349–383. MR**936086**, 10.1007/BF01404457**[W-2]**Hans Wenzl,*Quantum groups and subfactors of type 𝐵, 𝐶, and 𝐷*, Comm. Math. Phys.**133**(1990), no. 2, 383–432. MR**1090432****[W-3]**Hans Wenzl,*Braids and invariants of 3-manifolds*, Invent. Math.**114**(1993), no. 2, 235–275. MR**1240638**, 10.1007/BF01232670**[We]**H. Weyl,*The classical groups, their invariants and representations*, Princeton Univ. Press, 2nd ed., 1953. MR**1:42c (1st ed.)**

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Additional Information

**Juliana Erlijman**

Affiliation:
The Fields Institute, 222 College St., Toronto, Ontario M5T 3J1, Canada

Email:
jerlijma@fields.utoronto.ca

DOI:
https://doi.org/10.1090/S0002-9947-98-02007-8

Received by editor(s):
January 24, 1996

Article copyright:
© Copyright 1998
American Mathematical Society