Normal intermediate subfactors

Tamotsu TERUYA

doi:10.2969/jmsj/05020469

Tamotsu TERUYA

著者情報

キーワード: subfactor, Jones index, normal, Hopf algebra, Kac algebra, strongly outer action, modular lattice

ジャーナルフリー

1998 年 50 巻 2 号 p. 469-490

DOI https://doi.org/10.2969/jmsj/05020469

詳細

Published: 1998 Received: 1995/11/16 Available on J-STAGE: 2006/10/20 Accepted: - Advance online publication: - Revised: -
訂正情報
訂正日: 2006/10/20 訂正理由: - 訂正箇所: キーワード情報訂正内容: Right : subfactor, Jones index, normal, Hopf algebra, Kac algebra, strongly outer action, modular lattice

訂正日: 2006/10/20 訂正理由: - 訂正箇所: 引用文献情報訂正内容: Wrong : 1) D. Bisch, A note on intermediate subfactors, Pacific J. Math. 163 (1994), 201-216.
2) D. Bisch and U. Haagerup, Composition of subfactors: new examples of infinite depth subfactors, preprint.
3) M. Choda and H. Kosaki, Strongly outer actions for an inclusion of factors, J. Funct. Anal. 122 (1994), 315-332.
4) D. Evans and Y. Kawahigashi, Orbifold subfactors from Hecke algebras, Comm. Math. Phys. 165 (1994), 445-484.
5) F. Goodman, P, de la Harpe, and V. F. R. Jones, Coxeter graph and tower of algebras, vol. 14, MSRI Publ., Springer-Verlag, 1989.
6) S. Goto, Orbifold construction fro non-afd subfactor, Internat. J. Math. 5 (1994), 725-746.
7) M. Hall, Jr., The theory of groups, The Macmillan Company, New York, 1959.
8) M. Izumi, Application of fusion rules to classification of subfactors, Publ. RIMS, Kyoto Univ. 27 (1991), 953-994.
9) M. Izumi, Goldman's type theorem for index 3, Publ. RIMS, Kyoto Univ. 28 (1992), 833-843.
10) M. Izumi and Y. Kawahigashi, Classification of subfactors with the principal graph D⁽¹⁾_n, J. Funct. Anal. 112 (1993), 257-286.
11) V. F. R. Jones, Index for subfactors, Invent.math. 72 (1983), 1-25.
12) Y. Kawahigashi, On flatness of Ocneanu's connection on the dynkin diagrams and classification of subfactors, J. Funct. Anal. 127 (1995), 63-107.
13) H. Kosaki, Characterization of crossed product (properly infinite case), Pacific J. Math. 137 (1989), 159-167.
14) H. Kosaki, Automorphisms in the irreducible decomposition of sectors, Quantum and Non-Commutative Analysis, Kluwer Academic, 1993, pp. 305-316.
15) H. Kosaki and R. Longo, A remark on the minimal index of subfactors, J. Funct. Anal. 107 (1992), 458-470.
16) P. H. Loi, On the theory of index and type III factors, Ph.D. thesis, Pennsylvania State Univ., 1988.
17) R. Longo, Index of subfactors and statistics of quantum fields I, Commun. Math. Phys. 126 (1989), 217-247.
18) R. Longo, Index of subfactors and statistics of quantum fields II, Commun. Math. Phys. 130 (1990), 285-390.
19) S. Montgomery, Hopf algebras and their actions on rings, CBMS series number 82, 1992.
20) M. Nakamura and Z. Takeda, On the fundamental theorem of the Galois theory for finite factors, Japan Acad. 36 (1960), 313-318.
21) M. Nakamura and Z. Takeda, A Galois theory for finite factors, Proc.Japan Acad. 36 (1960), 258-260.
22) A. Ocneanu, Quantized groups, string algebras and galois theory for algebras, Operator Algebras and Applications, vo1.2, London Math. Soc. Lecture Note Series Vol.136, Cambridge Univ. Press, 1988, pp. 119-172.
23) A. Ocneanu, Quantum symmetry, differential geometry of finite graphs, and classification of subfactors, 1991, Univ. of Tokyo Seminary Notes.
24) P. Pimsner and S. Popa, Entropy and index for subfactors, Ann. Sci. Ecole Norm. Sup. 19 (1986), 57-106.
25) S. Popa, Correspondences, preprint.
26) S. Popa, Classification of subfactors: the reduction to commuting squares, Invent. Math. 101 (1990), 19-43.
27) S. Popa, Classification of amenable subfactors of type II, Acta Math. 172 (1994), 163-255.
28) T. Sano, Commuting co-commuting squares and finite dimensional Kac algebras, to appear in Pacific J. Math.
29) T. Sano and Y. Watatani, Angles between two subfactors, J. Operator Theory 32 (1994), 209-242.
30) Y. Sekine, Connections associated with finite-dimensional Kac algebra actions, Kyushu J. Math. 49 (1995), 253-269.
31) W. Szymanski, Finite index subfactors and Hopf algebra crossed products, Proc. Amer. Math. Soc. 120 (1994), 519-528.
32) T. Teruya, A characterization of normal extensions for subfactors, Proc. Amer. Math. Soc. 120 (1994), 781-783.
33) Y. Watatani, Lattices of intermediate subfactors, J. Funct. Anal., to appear.
34) S. Yamagami, A note on Ocneanu's approach to Jones' index theory, Internat. J. of Math. 4 (1993), 859-871.
35) S. Yamagami, Vector bundles and bimodules, Quantum and Non-Commutative Analysis, Kluwer Academic, 1993, pp. 321-329.
36) T. Yamanouchi, Construction of an outer action of a finite-dimensional Kac algebra on the AFD factor of type II₁, Inter. J. Math. 4 (1993), 1007-1045.

Right : [1] D. Bisch, A note on intermediate subfactors, Pacific J. Math. 163 (1994), 201-216.
[2] D. Bisch and U. Haagerup, Composition of subfactors: new examples of infinite depth subfactors, preprint.
[3] M. Choda and H. Kosaki, Strongly outer actions for an inclusion of factors, J. Funct. Anal. 122 (1994), 315-332.
[4] D. Evans and Y. Kawahigashi, Orbifold subfactors from Hecke algebras, Comm. Math. Phys. 165 (1994), 445-484.
[5] F. Goodman, P, de la Harpe, and V. F. R. Jones, Coxeter graph and tower of algebras, vol. 14, MSRI Publ., Springer-Verlag, 1989.
[6] S. Goto, Orbifold construction fro non-afd subfactor, Internat. J. Math. 5 (1994), 725-746.
[7] M. Hall, Jr., The theory of groups, The Macmillan Company, New York, 1959.
[8] M. Izumi, Application of fusion rules to classification of subfactors, Publ. RIMS, Kyoto Univ. 27 (1991), 953-994.
[9] M. Izumi, Goldman's type theorem for index 3, Publ. RIMS, Kyoto Univ. 28 (1992), 833-843.
[10] M. Izumi and Y. Kawahigashi, Classification of subfactors with the principal graph D⁽¹⁾_n, J. Funct. Anal. 112 (1993), 257-286.
[11] V. F. R. Jones, Index for subfactors, Invent. math. 72 (1983), 1-25.
[12] Y. Kawahigashi, On flatness of Ocneanu's connection on the dynkin diagrams and classification of subfactors, J. Funct. Anal. 127 (1995), 63-107.
[13] H. Kosaki, Characterization of crossed product (properly infinite case), Pacific J. Math. 137 (1989), 159-167.
[14] H. Kosaki, Automorphisms in the irreducible decomposition of sectors, Quantum and Non-Commutative Analysis, Kluwer Academic, 1993, pp. 305-316.
[15] H. Kosaki and R. Longo, A remark on the minimal index of subfactors, J. Funct. Anal. 107 (1992), 458-470.
[16] P. H. Loi, On the theory of index and type III factors, Ph. D. thesis, Pennsylvania State Univ., 1988.
[17] R. Longo, Index of subfactors and statistics of quantum fields I, Commun. Math. Phys. 126 (1989), 217-247.
[18] R. Longo, Index of subfactors and statistics of quantum fields II, Commun. Math. Phys. 130 (1990), 285-390.
[19] S. Montgomery, Hopf algebras and their actions on rings, CBMS series number 82, 1992.
[20] M. Nakamura and Z. Takeda, On the fundamental theorem of the Galois theory for finite factors, Japan Acad. 36 (1960), 313-318.
[21] M. Nakamura and Z. Takeda, A Galois theory for finite factors, Proc. Japan Acad. 36 (1960), 258-260.
[22] A. Ocneanu, Quantized groups, string algebras and galois theory for algebras, Operator Algebras and Applications, vol. 2, London Math. Soc. Lecture Note Series Vol. 136, Cambridge Univ. Press, 1988, pp. 119-172.
[23] A. Ocneanu, Quantum symmetry, differential geometry of finite graphs, and classification of subfactors, 1991, Univ. of Tokyo Seminary Notes.
[24] P. Pimsner and S. Popa, Entropy and index for subfactors, Ann. Sci. Ecole Norm. Sup. 19 (1986), 57-106.
[25] S. Popa, Correspondences, preprint.
[26] S. Popa, Classification of subfactors: the reduction to commuting squares, Invent. Math. 101 (1990), 19-43.
[27] S. Popa, Classification of amenable subfactors of type II, Acta Math. 172 (1994), 163-255.
[28] T. Sano, Commuting co-commuting squares and finite dimensional Kac algebras, to appear in Pacific J. Math.
[29] T. Sano and Y. Watatani, Angles between two subfactors, J. Operator Theory 32 (1994), 209-242.
[30] Y. Sekine, Connections associated with finite-dimensional Kac algebra actions, Kyushu J. Math. 49 (1995), 253-269.
[31] W. Szymanski, Finite index subfactors and Hopf algebra crossed products, Proc. Amer. Math. Soc. 120 (1994), 519-528.
[32] T. Teruya, A characterization of normal extensions for subfactors, Proc. Amer. Math. Soc. 120 (1994), 781-783.
[33] Y. Watatani, Lattices of intermediate subfactors, J. Funct. Anal., to appear.
[34] S. Yamagami, A note on Ocneanu's approach to Jones' index theory, Internat. J. of Math. 4 (1993), 859-871.
[35] S. Yamagami, Vector bundles and bimodules, Quantum and Non-Commutative Analysis, Kluwer Academic, 1993, pp. 321-329.
[36] T. Yamanouchi, Construction of an outer action of a finite-dimensional Kac algebra on the AFD factor of type II₁, Inter. J. Math. 4 (1993), 1007-1045.

訂正日: 2006/10/20 訂正理由: - 訂正箇所: PDFファイル訂正内容: -

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