Geodesic
$C^*$-basic construction between non-balanced quantum doubles
Czechoslovak Mathematical Journal, Tome 74 (2024) no. 2, pp. 611-621
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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.
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.
DOI : 10.21136/CMJ.2024.0022-24
Classification : 16S35, 46L05
Keywords: non-balanced quantum double; $C^*$-basic construction; crossed product; action 
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Xin, Qiaoling; Cao, Tianqing. $C^*$-basic construction between non-balanced quantum doubles. Czechoslovak Mathematical Journal, Tome 74 (2024) no. 2, pp. 611-621. doi: 10.21136/CMJ.2024.0022-24

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