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Abstract
In order to construct a class of new Turaev-braided group category with nontrivial associativity, the concept of a quasitriangular quasi-Turaev group coalgebras was recently introduced. Inside the definition, the conditions of invertibility of the R-matrix R and bijectivity of the antipode S are required. In this article, we prove that the antipode of a quasitriangular quasi-Turaev group coalgebra without the assumptions about invertibility of the antipode and R-matrix is inner, and a fortiori, bijective. As an application, we prove that for a quasitriangular quasi-Turaev group coalgebra, two conditions mentioned above are unnecessary.
Acknowledgments
We would like to thank Pukyong National University in Korea for its warm hospitality.
Additional information
Funding
Xiao-Li Fang was financially supported by the Natural Science Foundation of Zhejiang Province (No. LY14A010006) and the National Natural Science Foundation of China (No. 11571239). Tae-Hwa Kim was partly supported by a Research Grant of Pukyong National University (2017 Year). You-Qi Yin was partly supported by Zhejiang Education Department Project (No. Y201635981).