C * -basic construction between non-balanced quantum doubles

Qiaoling Xin; Tianqing Cao

Czechoslovak Mathematical Journal (2024)

  • Volume: 74, Issue: 2, page 611-621
  • ISSN: 0011-4642

Abstract

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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 ( ℂ X op ) * ⋊ 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.

How to cite

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Xin, Qiaoling, and Cao, Tianqing. "$C^*$-basic construction between non-balanced quantum doubles." Czechoslovak Mathematical Journal 74.2 (2024): 611-621. <http://eudml.org/doc/299392>.

@article{Xin2024,
abstract = {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 $(\mathbb \{C\}X^\{\rm op\})^*\rtimes \mathbb \{C\}G$. 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.},
author = {Xin, Qiaoling, Cao, Tianqing},
journal = {Czechoslovak Mathematical Journal},
keywords = {non-balanced quantum double; $C^*$-basic construction; crossed product; action},
language = {eng},
number = {2},
pages = {611-621},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {$C^*$-basic construction between non-balanced quantum doubles},
url = {http://eudml.org/doc/299392},
volume = {74},
year = {2024},
}

TY - JOUR
AU - Xin, Qiaoling
AU - Cao, Tianqing
TI - $C^*$-basic construction between non-balanced quantum doubles
JO - Czechoslovak Mathematical Journal
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 74
IS - 2
SP - 611
EP - 621
AB - 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 $(\mathbb {C}X^{\rm op})^*\rtimes \mathbb {C}G$. 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.
LA - eng
KW - non-balanced quantum double; $C^*$-basic construction; crossed product; action
UR - http://eudml.org/doc/299392
ER -

References

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