Hecke algebras of type \(A_ n\) and subfactors. (English) Zbl 0663.46055

Wenzl, Hans

Invent. Math. 92, No. 2, 349-383 (1988).

The author studies how subfactors of the hyperfinite \(II_ 1\) factor can be constructed using AF-algebras. He studies the invariants of the constructed subfactors, in particular its index in the sense of V. F. R. Jones [Invent. Math. 72, 1-25 (1983; Zbl 0508.46040)], and gives an upper bound for the size of its centralizer. These results are then applied to the series of complex Hecke algebras \(H_ n(q)\) of type \(A_{n-1}\) equipped with a suitable trace.

Reviewer: G.Loupias

MSC:

46L35

Classifications of \(C^*\)-algebras

Keywords:

Jones index; subfactors of the hyperfinite \(II_ 1\) factor; AF-algebras; centralizer; complex Hecke algebras; trace

Citations:

Zbl 0508.46040

References:

[1]	[B] Birman, J.: Braids, links and mapping class groups. Ann. Math. Stud.82 (1974)
[2]	[F] [FYHLMO] Freyd, P., Yetter, D., Hoste, J., Lickorish, W.B.R., Millett, K., Ocneanu, A.: A new polynomial invariant of knots and links. Bull. Am. Math. Soc.12, 239-246 (1985) · Zbl 0572.57002
[3]	[G] Gantmacher, F.R.: Matrix theory, Vol. II, Chelsea Publishing Company: New York, 1974
[4]	[HW] Handelman, D., Wenzl, H.: Closedness of index values for subfactors. Proc. Am Math. Soc.101, 277-282 (1987) · Zbl 0638.46042
[5]	[H] Hoefsmit, P.N.: Representations of Hecke algebras of finite groups with BN pairs of classical type. Thesis, University of British Columbia, 1974
[6]	[JK] James, G., Kerber, A.: The representations of the symmetric group. Addison-Wesley: Reading, Mass
[7]	[J-1] Jones, V.F.R.: Index for subfactors. Invent. Math.72, 1-25 (1983) · Zbl 0508.46040
[8]	[J-2] Jones, V.F.R.: Braid groups, Hecke algebras and type II1 factors. Japan-US Conference proceedings, 1983
[9]	[J-3] Jones, V.F.R.: A polynomial invariant for knots via von Neumann algebras. Bull. Am. Math. Soc.12, 103-111 (1985) · Zbl 0564.57006
[10]	[J-4] Jones, V.F.R.: Hecke algebra representations of braid groups. Ann. Math.126, 335-388 (1987) · Zbl 0631.57005
[11]	[L] Littlewood, D.: The theory of group characters. Oxford University Press, 1940 · JFM 66.0093.02
[12]	[Mc] McDonald, I.: Symmetric functions and Hall polynomials. Clarendon Press: Oxford, 1979
[13]	[M] Murnaghan, F.: The theory of group representations. Dover, New York, 1939
[14]	[O] Ocneanu, A.: A polynomial invariant for knots: A combinatorial and an algebraic approach (Preprint, see [FYHLMO] for a shortened version)
[15]	[PP1] Pimsner, M., Popa, S.: Entropy and index for subfactors. Ann. Sci. Ec. Norm. Super, 4 e serie,19, 57-106 (1986) · Zbl 0646.46057
[16]	[PP2] Pimsner, M., Popa, S.: Iterating the basic construction (Preprint, INCREST) · Zbl 0706.46047
[17]	[St] Størmer, E.: Symmetric states of infinite tensor products ofC * algebras J. Funct. Anal.3, 48-68 (1969) · Zbl 0167.43403
[18]	[SV] Stratila, S., Voiculescu, D.: Representations ofAF-algebras and of the groupU(?). Lect. Notes Math., Vol. 486. Berlin Heidelberg New York: Springer 1975
[19]	[T] Thoma, E.: Die unzerlegbaren, positiv-definiten Klassenfunktionen der abzaehlbaren, unendlichen, symmetrischen Gruppe. Math. Z.85, 40-61 (1964) · Zbl 0192.12402
[20]	[Wa] Wassermann, A.: Automorphic actions of compact groups on operator algebras. Thesis, University of Pennsylvania 1981
[21]	[W-1] Wenzl, H.: On sequences of projections. C.R. Math. Rep. Acad. Sci. Canada IX,1, 5-9 (1987) · Zbl 0622.47019
[22]	[W-2] Wenzl, H.: Representations of Hecke algebras and subfactors. Thesis, University of Pennsylvania 1985