Weak Hopf algebras and singular solutions of Quantum Yang-Baxter equation

Abstract

We investigate a generalization of Hopf algebra ${\mathfrak{sl}}_q(2)$ by weakening the invertibility of the generator $K$, i.e. exchanging its invertibility $K{K}^{-1}=1$ to the regularity $K\overline{K}K=K$. This leads to a weak Hopf algebra $w{\mathfrak{sl}}_q(2)$ and a $J$-weak Hopf algebra $v{\mathfrak{sl}}_q(2)$ which are studied in detail. It is shown that the monoids of group-like elements of $w{\mathfrak{sl}}_q(2)$ and $v{\mathfrak{sl}}_q(2)$ are regular monoids, which supports the general conjucture on the connection betweek weak Hopf algebras and regular monoids. Moreover, from $w{\mathfrak{sl}}_q(2)$ a quasi-braided weak Hopf algebra ${\overline{U}}_q^w$ is constructed and it is shown that the corresponding quasi-$R$-matrix is regular $R^w{\hat{R}}^w{R}^w=R^w$.

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