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#### Quantum Algebramathscidoc:1702.29001

Communications in Mathematical Physcis, 225, 27, 2002
We investigate a generalization of Hopf algebra $\mathfrak{sl}_{q}\left( 2\right)$ by weakening the invertibility of the generator $K$, i.e. exchanging its invertibility $KK^{-1}=1$ to the regularity $K\overline{K}K=K$. This leads to a weak Hopf algebra $w\mathfrak{sl}_{q}\left( 2\right)$ and a $J$-weak Hopf algebra $v\mathfrak{sl}_{q}\left( 2\right)$ which are studied in detail. It is shown that the monoids of group-like elements of $w\mathfrak{sl}_{q}\left( 2\right)$ and $v\mathfrak{sl}_{q}\left( 2\right)$ are regular monoids, which supports the general conjucture on the connection betweek weak Hopf algebras and regular monoids. Moreover, from $w\mathfrak{sl}_{q}\left( 2\right)$ a quasi-braided weak Hopf algebra $\overline{U}_{q}^{w}$ is constructed and it is shown that the corresponding quasi-$R$-matrix is regular $R^{w}\hat{R}^{w}R^{w}=R^{w}$.
weak Hopf algebra; quantum Yang-Baxter equation; singular solution
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@inproceedings{fang2002weak,
title={Weak Hopf Algebras and Singular Solutions of Quantum Yang–Baxter Equation},
author={Fang Li, and Steven Duplij},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170209100548424452370},
booktitle={Communications in Mathematical Physcis},
volume={225},
pages={27},
year={2002},
}

Fang Li, and Steven Duplij. Weak Hopf Algebras and Singular Solutions of Quantum Yang–Baxter Equation. 2002. Vol. 225. In Communications in Mathematical Physcis. pp.27. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170209100548424452370.
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