Towards a quantum Galois theory for quantum double algebras of finite groups

Lining, Jiang

doi:10.1090/S0002-9939-10-10315-3

Towards a quantum Galois theory for quantum double algebras of finite groups
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Proc. Amer. Math. Soc. 138 (2010), 2793-2801 Request permission

Abstract:

Suppose that $G$ is a finite group and $D(G)$ the quantum double algebra of $G$. Let $\mathcal A$ be the field algebra of $G$-spin models. There is a natural action of $D(G)$ on $\mathcal A$ such that $\mathcal A$ becomes a $D(G)$-module algebra. For a subgroup $H$ of $G$, there is a Hopf subalgebra $D(G;H)$ of $D(G)$. Based on the concrete construction of a $D(G;H)$ fixed point subalgebra, the paper proves that $D(G;H)$ is Galois closed and thus gives a quantum Galois theory in the field algebra of $G$-spin models.

References

Eiichi Abe, Hopf algebras, Cambridge Tracts in Mathematics, vol. 74, Cambridge University Press, Cambridge-New York, 1980. Translated from the Japanese by Hisae Kinoshita and Hiroko Tanaka. MR 594432
Jeffrey Bergen, A correspondence theorem for modules over Hopf algebras, Proc. Amer. Math. Soc. 121 (1994), no. 2, 343–345. MR 1211578, DOI 10.1090/S0002-9939-1994-1211578-X
S. U. Chase, D. K. Harrison, and Alex Rosenberg, Galois theory and Galois cohomology of commutative rings, Mem. Amer. Math. Soc. 52 (1965), 15–33. MR 195922
Stephen U. Chase and Moss E. Sweedler, Hopf algebras and Galois theory, Lecture Notes in Mathematics, Vol. 97, Springer-Verlag, Berlin-New York, 1969. MR 0260724
K. A. Dancer, P. S. Isac, and J. Links, Representations of the quantum doubles of finite group algebras and spectral parameter dependent solutions of the Yang-Baxter equation, J. Math. Phys. 47 (2006), no. 10, 103511, 18. MR 2268877, DOI 10.1063/1.2359575
Chongying Dong and Geoffrey Mason, On quantum Galois theory, Duke Math. J. 86 (1997), no. 2, 305–321. MR 1430435, DOI 10.1215/S0012-7094-97-08609-9
V. O. Ferreira, L. S. I. Murakami, and A. Paques, A Hopf-Galois correspondence for free algebras, J. Algebra 276 (2004), no. 1, 407–416. MR 2054404, DOI 10.1016/S0021-8693(03)00502-7
M. D. Gould, Quantum double finite group algebras and their representations, Bull. Austral. Math. Soc. 48 (1993), no. 2, 275–301. MR 1238802, DOI 10.1017/S0004972700015707
Akihide Hanaki, Masahiko Miyamoto, and Daisuke Tambara, Quantum Galois theory for finite groups, Duke Math. J. 97 (1999), no. 3, 541–544. MR 1682988, DOI 10.1215/S0012-7094-99-09720-X
Li Ning Jiang, The duality theorem in field algebra of $G$-spin model, Acta Math. Sinica (Chinese Ser.) 45 (2002), no. 1, 37–42 (Chinese, with English and Chinese summaries). MR 1923662
Vaughan F. R. Jones, Subfactors and knots, CBMS Regional Conference Series in Mathematics, vol. 80, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1991. MR 1134131, DOI 10.1090/cbms/080
Yasuyuki Kawahigashi, Quantum Galois correspondence for subfactors, J. Funct. Anal. 167 (1999), no. 2, 481–497. MR 1716205, DOI 10.1006/jfan.1999.3453
H. F. Kreimer and M. Takeuchi, Hopf algebras and Galois extensions of an algebra, Indiana Univ. Math. J. 30 (1981), no. 5, 675–692. MR 625597, DOI 10.1512/iumj.1981.30.30052
Geoffrey Mason, The quantum double of a finite group and its role in conformal field theory, Groups ’93 Galway/St. Andrews, Vol. 2, London Math. Soc. Lecture Note Ser., vol. 212, Cambridge Univ. Press, Cambridge, 1995, pp. 405–417. MR 1337285, DOI 10.1017/CBO9780511629297.009
F. Van Oystaeyen and Y. Zhang, Galois-type correspondences for Hopf-Galois extensions, Proceedings of Conference on Algebraic Geometry and Ring Theory in honor of Michael Artin, Part III (Antwerp, 1992), 1994, pp. 257–269. MR 1291021, DOI 10.1007/BF00960864
David E. Radford, Minimal quasitriangular Hopf algebras, J. Algebra 157 (1993), no. 2, 285–315. MR 1220770, DOI 10.1006/jabr.1993.1102
K. Szlachányi and P. Vecsernyés, Quantum symmetry and braid group statistics in $G$-spin models, Comm. Math. Phys. 156 (1993), no. 1, 127–168. MR 1234107
A. Van Daele and Y. H. Zhang, Galois theory for multiplier Hopf algebras with integrals, Algebr. Represent. Theory 2 (1999), no. 1, 83–106. MR 1688472, DOI 10.1023/A:1009938708033
Tadashi Yanai, Galois correspondence theorem for Hopf algebra actions, Algebraic structures and their representations, Contemp. Math., vol. 376, Amer. Math. Soc., Providence, RI, 2005, pp. 393–411. MR 2147038, DOI 10.1090/conm/376/06974
S. C. Zhang, Y. Z. Zhang, Hopf Galois extension braided tensor categories, arXiv:math.RA/0309448, 7, 19, April 2006.