# The basic construction from the conditional expectation on the quantum double of a finite group

Qiaoling Xin; Lining Jiang; Zhenhua Ma

Czechoslovak Mathematical Journal (2015)

- Volume: 65, Issue: 2, page 347-359
- ISSN: 0011-4642

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topXin, Qiaoling, Jiang, Lining, and Ma, Zhenhua. "The basic construction from the conditional expectation on the quantum double of a finite group." Czechoslovak Mathematical Journal 65.2 (2015): 347-359. <http://eudml.org/doc/270102>.

@article{Xin2015,

abstract = {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 $\mathbb \{C\}H$ (or $\mathbb \{C\}G$) 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 $C(G/H\times G)\rtimes \mathbb \{C\}G$, and proves that there is an algebra isomorphism between $\langle D(G),e\rangle $ and $C(G/H\times G)\rtimes \mathbb \{C\}G$.},

author = {Xin, Qiaoling, Jiang, Lining, Ma, Zhenhua},

journal = {Czechoslovak Mathematical Journal},

keywords = {conditional expectation; basic construction; quantum double; quasi-basis},

language = {eng},

number = {2},

pages = {347-359},

publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},

title = {The basic construction from the conditional expectation on the quantum double of a finite group},

url = {http://eudml.org/doc/270102},

volume = {65},

year = {2015},

}

TY - JOUR

AU - Xin, Qiaoling

AU - Jiang, Lining

AU - Ma, Zhenhua

TI - The basic construction from the conditional expectation on the quantum double of a finite group

JO - Czechoslovak Mathematical Journal

PY - 2015

PB - Institute of Mathematics, Academy of Sciences of the Czech Republic

VL - 65

IS - 2

SP - 347

EP - 359

AB - 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 $\mathbb {C}H$ (or $\mathbb {C}G$) 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 $C(G/H\times G)\rtimes \mathbb {C}G$, and proves that there is an algebra isomorphism between $\langle D(G),e\rangle $ and $C(G/H\times G)\rtimes \mathbb {C}G$.

LA - eng

KW - conditional expectation; basic construction; quantum double; quasi-basis

UR - http://eudml.org/doc/270102

ER -

## References

top- Bántay, P., 10.1016/0370-2693(90)90676-W, Phys. Lett., B 245 (1990), 477-479. (1990) MR1070067DOI10.1016/0370-2693(90)90676-W
- 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). (1987) Zbl0905.46046MR0887100
- Dancer, K. A., Isac, P. S., Links, J., 10.1063/1.2359575, J. Math. Phys. 47 (2006), 103511, 18 pages. (2006) Zbl1112.17016MR2268877DOI10.1063/1.2359575
- Jiang, L., 10.1090/S0002-9939-10-10315-3, Proc. Am. Math. Soc. 138 (2010), 2793-2801. (2010) Zbl1215.16020MR2644893DOI10.1090/S0002-9939-10-10315-3
- Jiang, L., 10.1360/03YS0119, Sci. China, Ser. A 48 (2005), 57-66. (2005) Zbl1177.82024MR2156615DOI10.1360/03YS0119
- Jiang, L., Zhu, G., ${C}^{*}$-index in double algebra of finite group, Trans. Beijing Inst. Technol. 23 (2003), 147-148 Chinese. (2003) Zbl1084.46044MR1976172
- 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). (1991) Zbl0743.46058MR1134131
- Jones, V. F. R., 10.1007/BF01389127, Invent. Math. 72 (1983), 1-25. (1983) Zbl0508.46040MR0696688DOI10.1007/BF01389127
- Kassel, C., Quantum Groups, Graduate Texts in Mathematics 155 Springer, Berlin (1995). (1995) Zbl0808.17003MR1321145
- Kawahigashi, Y., Longo, R., 10.4007/annals.2004.160.493, Ann. Math. 160 (2004), 493-522. (2004) Zbl1083.46038MR2123931DOI10.4007/annals.2004.160.493
- Kosaki, H., 10.1016/0022-1236(86)90085-6, J. Funct. Anal. 66 (1986), 123-140. (1986) MR0829381DOI10.1016/0022-1236(86)90085-6
- Longo, R., 10.1007/BF02473354, Commun. Math. Phys. 130 (1990), 285-309. (1990) Zbl0705.46038MR1059320DOI10.1007/BF02473354
- Longo, R., 10.1007/BF02125124, Commun. Math. Phys. 126 (1989), 217-247. (1989) Zbl0682.46045MR1027496DOI10.1007/BF02125124
- Radford, D. E., 10.1006/jabr.1993.1102, J. Algebra 157 (1993), 285-315. (1993) Zbl0787.16028MR1220770DOI10.1006/jabr.1993.1102
- Sweedler, M. E., Hopf Algebras, Mathematics Lecture Note Series W. A. Benjamin, New York (1969). (1969) Zbl0203.31601MR0252485
- Watatani, Y., Index for ${C}^{*}$-subalgebras, Mem. Am. Math. Soc. 83 (1990). (1990) MR0996807

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