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The basic construction from the conditional expectation on the quantum double of a finite group
Czechoslovak Mathematical Journal, Tome 65 (2015) no. 2, pp. 347-359 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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Let $G$ be a finite group and $H$ a subgroup. Denote by $D(G;H)$ (or $D(G)$) the crossed product of $C(G)$ and $\Bbb {C}H$ (or $\Bbb {C}G$) with respect to the adjoint action of the latter on the former. Consider the algebra $\langle D(G), e\rangle $ generated by $D(G)$ and $e$, where we regard $E$ as an idempotent operator $e$ on $D(G)$ for a certain conditional expectation $E$ of $D(G)$ onto $D(G;H)$. Let us call $\langle D(G), e\rangle $ the basic construction from the conditional expectation $E\colon D(G)\rightarrow D(G;H)$. The paper constructs a crossed product algebra $C(G/H\times G)\rtimes \Bbb {C}G$, and proves that there is an algebra isomorphism between $\langle D(G),e\rangle $ and $C(G/H\times G)\rtimes \Bbb {C}G$.
Let $G$ be a finite group and $H$ a subgroup. Denote by $D(G;H)$ (or $D(G)$) the crossed product of $C(G)$ and $\Bbb {C}H$ (or $\Bbb {C}G$) with respect to the adjoint action of the latter on the former. Consider the algebra $\langle D(G), e\rangle $ generated by $D(G)$ and $e$, where we regard $E$ as an idempotent operator $e$ on $D(G)$ for a certain conditional expectation $E$ of $D(G)$ onto $D(G;H)$. Let us call $\langle D(G), e\rangle $ the basic construction from the conditional expectation $E\colon D(G)\rightarrow D(G;H)$. The paper constructs a crossed product algebra $C(G/H\times G)\rtimes \Bbb {C}G$, and proves that there is an algebra isomorphism between $\langle D(G),e\rangle $ and $C(G/H\times G)\rtimes \Bbb {C}G$.
DOI : 10.1007/s10587-015-0179-0
Classification : 16S35, 16S99, 16W22, 20D99
Keywords: conditional expectation; basic construction; quantum double; quasi-basis 
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Xin, Qiaoling; Jiang, Lining; Ma, Zhenhua. The basic construction from the conditional expectation on the quantum double of a finite group. Czechoslovak Mathematical Journal, Tome 65 (2015) no. 2, pp. 347-359. doi: 10.1007/s10587-015-0179-0

[1] Bántay, P.: Orbifolds and Hopf algebras. Phys. Lett., B 245 (1990), 477-479. | DOI | MR

[2] Bratteli, O., Robinson, D. W.: Operator Algebras and Quantum Statistical Mechanics. 1. $C^*$- and $W^*$-Algebras, Symmetry Groups, Decomposition of States. Texts and Monographs in Physics Springer, New York (1987). | MR | Zbl

[3] Dancer, K. A., Isac, P. S., Links, J.: Representations of the quantum doubles of finite group algebras and spectral parameter dependent solutions of the Yang-Baxter equation. J. Math. Phys. 47 (2006), 103511, 18 pages. | DOI | MR | Zbl

[4] Jiang, L.: Towards a quantum Galois theory for quantum double algebras of finite groups. Proc. Am. Math. Soc. 138 (2010), 2793-2801. | DOI | MR | Zbl

[5] Jiang, L.: $C^*$-index of observable algebras in $G$-spin model. Sci. China, Ser. A 48 (2005), 57-66. | DOI | MR | Zbl

[6] Jiang, L., Zhu, G.: $C^*$-index in double algebra of finite group. Trans. Beijing Inst. Technol. 23 (2003), 147-148 Chinese. | MR | Zbl

[7] Jones, V. F. R.: Subfactors and Knots. Expository lectures from the CBMS regional conference, Annapolis, USA, 1988. Regional Conference Series in Mathematics 80 AMS, Providence (1991). | MR | Zbl

[8] Jones, V. F. R.: Index for subfactors. Invent. Math. 72 (1983), 1-25. | DOI | MR | Zbl

[9] Kassel, C.: Quantum Groups. Graduate Texts in Mathematics 155 Springer, Berlin (1995). | MR | Zbl

[10] Kawahigashi, Y., Longo, R.: Classification of local conformal nets. Case $c<1$. Ann. Math. 160 (2004), 493-522. | DOI | MR | Zbl

[11] Kosaki, H.: Extension of Jones' theory on index to arbitrary factors. J. Funct. Anal. 66 (1986), 123-140. | DOI | MR

[12] 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 | MR | Zbl

[13] Longo, R.: Index of subfactors and statistics of quantum fields. I. Commun. Math. Phys. 126 (1989), 217-247. | DOI | MR | Zbl

[14] Radford, D. E.: Minimal quasitriangular Hopf algebras. J. Algebra 157 (1993), 285-315. | DOI | MR | Zbl

[15] Sweedler, M. E.: Hopf Algebras. Mathematics Lecture Note Series W. A. Benjamin, New York (1969). | MR | Zbl

[16] Watatani, Y.: Index for $C^*$-subalgebras. Mem. Am. Math. Soc. 83 (1990). | MR

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