Article
Full entry | Fulltext not available
(moving wall
24 months)
Feedback
Keywords:
non-balanced quantum double; $C^*$-basic construction; crossed product; action
non-balanced quantum double; $C^*$-basic construction; crossed product; action
Summary:
For finite groups $X$, $G$ and the right $G$-action on $X$ by group automorphisms, the non-balanced quantum double $D(X;G)$ is defined as the crossed product $(\Bbb {C}X^{\rm op})^*\rtimes \Bbb {C}G$. We firstly prove that $D(X;G)$ is a finite-dimensional Hopf $C^*$-algebra. For any subgroup $H$ of $G$, $D(X;H)$ can be defined as a Hopf $C^*$-subalgebra of $D(X;G)$ in the natural way. Then there is a conditonal expectation from $D(X;G)$ onto $D(X;H)$ and the index is $[G;H]$. Moreover, we prove that an associated natural inclusion of non-balanced quantum doubles is the crossed product by the group algebra.
References:
[1] Bratteli, O., Robinson, D. W.: Operator Algebras and Quantum Statistical Mechanics. Vol. 1. C*- and W*-Algebras. Symmetry Groups. Decomposition of States. Texts and Monographs in Physics. Springer, New York (1987). DOI 10.1007/978-3-662-02520-8 | MR 0887100 | Zbl 0905.46046
[2] Cao, T., Xin, Q., Wei, X., Jiang, L.: Jones type basic construction on Hopf spin models. Mathematics 8 (2020), Article ID 1547, 9 pages. DOI 10.3390/math8091547
[3] Chen, L., Li, F.: On the structure and representations of non-balanced quantum doubles. Appl. Math., Ser. B (Engl. Ed.) 27 (2012), 475-488. DOI 10.1007/s11766-012-2852-5 | MR 3001093 | Zbl 1289.16049
[4] De, S., Kodiyalam, V.: Note on infinite iterated crossed products of Hopf algebras and the Drinfeld double. J. Pure Appl. Algerba 219 (2015), 5305-5313. DOI 10.1016/j.jpaa.2015.05.013 | MR 3390022 | Zbl 1325.16028
[5] Drinfel'd, V. G.: Quantum groups. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 AMS, Providence (1987), 798-820. MR 0934283 | Zbl 0667.16003
[6] Hiai, F.: Minimizing indices of conditional expectations onto a subfactor. Publ. Res. Inst. Math. Sci. 24 (1988), 673-678. DOI 10.2977/PRIMS/1195174872 | MR 0976765 | Zbl 0679.46050
[7] Jiang, L.: $C^*$-index of observable algebras in $G$-spin model. Sci. China, Ser. A 48 (2005), 57-66. DOI 10.1360/03ys0119 | MR 2156615 | Zbl 1177.82024
[8] Jiang, L., Zhu, G.: $C^*$-index in double algebra of finite group. Trans. Beijing Inst. Technol. 23 (2003), 147-148 Chinese. MR 1976172 | Zbl 1084.46044
[9] Kassel, C.: Quantum Groups. Graduate Texts in Mathematics 155. Springer, New York (1995). DOI 10.1007/978-1-4612-0783-2 | MR 1321145 | Zbl 0808.17003
[10] Kawahigashi, Y., Longo, R.: Classification of local conformal nets. Case $c<1$. Ann. Math. (2) 160 (2004), 493-522. DOI 10.4007/annals.2004.160.493 | MR 2123931 | Zbl 1083.46038
[11] Kosaki, H.: Extension of Jones' theory on index to arbitrary factors. J. Funct. Anal. 66 (1986), 123-140. DOI 10.1016/0022-1236(86)90085-6 | MR 0829381 | Zbl 0607.46034
[12] Loi, P. H.: Sur la théorie de l'indice et les facteurs de type III. C. R. Acad. Sci., Paris, Sér. 305 (1987), 423-426 French. MR 0916344 | Zbl 0622.46042
[13] Longo, R.: Index of subfactors and statistics of quantum fields. II. Correspondences, braid group statistics and Jones polynomial. Commun. Math. Phys. 130 (1990), 285-309. DOI 10.1007/BF02473354 | MR 1059320 | Zbl 0705.46038
[14] Montgomery, S.: Hopf Algebras and Their Actions on Rings. Regional Conference Series in Mathematics 82. AMS, Providence (1993). DOI 10.1090/cbms/082 | MR 1243637 | Zbl 0793.16029
[15] Murphy, G. J.: $C^*$-Algebras and Operator Theory. Academic Press, Boston (1990). DOI 10.1016/c2009-0-22289-6 | MR 1074574 | Zbl 0714.46041
[16] Pimsner, M., Popa, S.: Entropy and index for subfactors. Ann. Sci. Éc. Norm. Supér. (4) 19 (1986), 57-106. DOI 10.24033/asens.1504 | MR 0860811 | Zbl 0646.46057
[17] Radford, D. E.: Minimal quasitriangular Hopf algebras. J. Algebra 157 (1993), 285-315. DOI 10.1006/jabr.1993.1102 | MR 1220770 | Zbl 0787.16028
[18] Sweedler, M. E.: Hopf Algebras. W. A. Benjamin, New York (1969). MR 0252485 | Zbl 0194.32901
[19] Vaes, S., Daele, A. Van: Hopf $C^*$-algebras. Proc. Lond. Math. Soc., III. Ser. 82 (2001), 337-384. DOI 10.1112/S002461150101276X | MR 1806875 | Zbl 1028.46100
[20] Watatani, Y.: Index for $C^*$-subalgebras. Mem. Am. Math. Soc. 424 (1990), 117 pages. DOI 10.1090/memo/0424 | MR 0996807 | Zbl 0697.46024
[21] Xin, Q., Jiang, L.: Symmetric structure of field algebra of $G$-spin models determined by a normal subgroup. J. Math. Phys. 55 (2014), Article ID 091703, 9 pages. DOI 10.1063/1.4896327 | MR 3390774 | Zbl 1442.81056
[22] Xin, Q., Jiang, L., Cao, T.: $C^*$-basic construction from the conditional expectation on the Drinfeld double. J. Funct. Spaces 2019 (2019), Article ID 2041079, 7 pages. DOI 10.1155/2019/2041079 | MR 3976604 | Zbl 1436.46051
Search
Partner of
