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

Journal of Mathematical Physics

2014

Let G be a finite group and H a normal subgroup. D(H; G) is the crossed product of C(H) and \documentclass[12pt]{minimal}\begin{document}${\mathbb {C}}G$\end{document}CG which is only a subalgebra of D(G), the double algebra of G. One can construct a C*-subalgebra \documentclass[12pt]{minimal}\begin{document}${\mathcal {F}}_H$\end{document}FH of the field algebra \documentclass[12pt]{minimal}\begin{document}$\mathcal {F}$\end{document}F of G-spin models, so that \documentclass[12pt]{minimal}\begin{document}${\mathcal {F}}_H$\end{document}FH is a D(H; G)-module algebra, whereas \documentclass[12pt]{minimal}\begin{document}$\mathcal {F}$\end{document}F is not. Then the observable algebra \documentclass[12pt]{minimal}\begin{document}${\mathcal {A}}_{(H,G)}$\end{document}A(H,G) is obtained as the D(H; G)-invariant subalgebra of \documentclass[12pt]{minimal}\begin{document}${\mathcal {F}}_H$\end{document}FH, and there exists a unique C*-representation of D(H; G) such that D(H; G) and \documentclass[12pt]{minimal}\begin{document}${\mathcal {A}}_{(H,G)}$\end{document}A(H,G) are commutants with each other.

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“…(1) Different from the case of D(G; H), the crossed product of C(G) and C H ,8,11 D(H; G) is not a Hopf subalgebra of D(G), even though it is a subalgebra of D(G).…”

mentioning

confidence: 99%

Symmetric structure of field algebra of G-spin models determined by a normal subgroup

Journal of Mathematical Physics

2014

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C^∗‐index of observable algebra in the field algebra determined by a normal group

Cao

Math Methods in App Sciences

2021

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Let G be a finite group and H a normal subgroup. By D(H; G), we denote the crossed product of C(H) and CG, which is only a subalgebra of the quantum double D(G) of G. One can construct a C * -subalgebra  H of the field algebra  of G-spin models, such that  H is a D(H; G)-module algebra. The concrete construction of D(H; G)-invariant subalgebra  (H,G) of  H is given. Moreover, the C * -index of the conditional expectation z H from  H onto  (H,G) is calculated in terms of the quasi-basis for z H .

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

2015

Czech Math J