[1] Abe E. Hopf Algebras.Cambridge: Cambridge University Press, 2004 [2] Blanchard E.On finiteness of the N-dimensional Hopf ${C}^*$-algebras. Operator theoretical methods (Timisoara, 1998). Theta Found, 2000: 39-46 [3] Bratteli O. Inductive limits of finite dimensional C$^*$-algebras. Trans Amer Math Soc, 1972, 171: 195-234 [4] Jeong J A, Park G H. Saturated actions by finite-dimensional Hopf $*$-algebras on ${C}^*$-algebras. International Journal of Mathematics, 2008, 19(2): 125-144 [5] Jones V F R. Index for subfactors. Inventiones Mathematicae, 1983, 72(1): 1-25 [6] Jones V F R. Hecke algebra representations of braid groups and link polynomials. Ann Math, 1987, 126: 335-388 [7] Kajiwara T, Pinzari C, Watatani Y. Jones index theory for Hilbert ${C}^*$-bimodules and its equivalence with conjugation theory. J Funct Anal, 2004, 215(1): 1-49 [8] Kajiwara T, Watatani Y. Jones index theory by Hilbert ${C}^*$-bimodules and K-theory. Trans Amer Math Soc, 2000, 352(8): 3429-3472 [9] Kauffman L H. State models and the Jones polynomial. Topology, 1987, 26(3): 395-407 [10] Kosaki H. Extension of Jones' theory on index to arbitrary factors. J Funct Anal, 1986, 66(1): 123-140 [11] Lance E C.Hilbert ${C}^*$-modules: a Toolkit for Operator Algebraists. Cambridge: Cambridge University Press, 1995 [12] Longo R. Index of subfactors and statistics of quantum fields. Ⅰ. Commun Math Phys, 1989, 126(2): 217-247 [13] Longo R. Index of subfactors and statistics of quantum fields. Ⅱ. Correspondences, braid group statistics and Jones polynomial. Commun Math Phys, 1990, 130(2): 285-309 [14] Manuilov V M, Troitsky E V.Hilbert ${C}^*$-modules. Providence: American Mathematical Society, 2005 [15] Ng C K. Duality of Hopf ${C}^*$-algebras. International Journal of Mathematics, 2002, 13(9): 1009-1025 [16] Ng C K. Morita equivalences between fixed point algebras and crossed products. Mathematical Proceedings of the Cambridge Philosophical Society, 1999, 125(1): 43-52 [17] Nill F, Szlachányi K. Quantum chains of Hopf algebras with quantum double cosymmetry. Commun Math Phys, 1997, 187(1): 159-200 [18] Pimsner M, Popa S. Entropy and index for subfactors. Annales Scientifiques de l'École Normale Supérieure, 1986, 19(1): 57-106 [19] Serre J-P.Linear Representations of Finite Groups. Graduate Texts in Mathematics, Vol 42. New York, Heidelberg: Springer-Verlag, 1977 [20] Szlachányi K, Vecsernyés P. Quantum symmetry and braid group statistics in $G$-spin models. Commun Math Phys, 1993, 156(1): 127-168 [21] Szymański W, Peligrad C. Saturated actions of finite dimensional Hopf $*$-algebras on ${C}^*$-algebras. Mathematica Scandinavica, 1994, 75: 219-239 [22] Takai H. On a duality for crossed products of ${C}^*$-algebras. J Funct Anal, 1975, 19(1): 25-39 [23] Takesaki M.Theory of Operator Algebra Ⅰ. Berlin, Heidelberg, New York: Springer-Verlag, 2002 [24] Temperley N, Lieb E. Relations between the 'percolation' and 'colouring' problem and other graph-theoretical problems associated with regular planar lattices: some exact results for the ‘percolation' problem. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 1971, 322(1549): 251-280 [25] Vaes S, Van Daele A. Hopf ${C}^*$-algebras. Proceedings of the London Mathematical Society, 2001, 82(3): 337-384 [26] Van Daele A. The Haar measure on finite quantum groups. Proceedings of the American Mathematical Society, 1997, 125(12): 3489-3500 [27] Watatani Y. Index for ${C}^*$-subalgebras-introduction. Memoirs of the American Mathematical Society, 1990, 83(424): 1-117 [28] Wei X M, Jiang L N, Xin Q L. The structure of the observable algebra determined by a Hopf $*$-subalgebra in Hopf spin models. Filomat, 2021, 35(2): 485-500 [29] Wei X M, Jiang L N, Xin Q L. The field algebra in Hopf spin models determined by a Hopf $*$-subalgebra and its symmetric structure. Acta Mathematica Scientia, 2021, 41B(3): 907-924 [30] Wei X M, Jiang L N. The ${C}^*$-algebra index for observable algebra in non-equilibrium Hopf spin models. Annals of Functional Analysis, 2022, 13: 73 [31] Woronowicz S L. Compact matrix pseudogroups. Commun Math Phys, 1987, 111(4): 613-665 [32] Xin Q L, Cao T Q, Jiang L N. ${C}^*$-index of observable algebra in the field algebra determined by a normal group. Mathematical Methods in the Applied Sciences, 2022, 45(7): 3689-3697 |