Weak Hopf algebras and some new solutions of the quantum Yang-Baxter equation
A bialgebra \(H\) is called a weak Hopf algebra if there is a linear map \(T\) from \(H\) to \(H\) such that \(I*T*I=I\) and \(T*I*T=T\), \(*\) the convolution product. This generalizes the definition of a left Hopf algebra as studied by \textit{J. A. Green, W. D. Nichols} and the reviewer [J. Algebra 65, 399-411 (1980; Zbl 0439.16008)]. The author also defines a quasi-braided almost bialgebra. For a weak Hopf algebra, he constructs a quantum quasi-double which is a quasi-braided almost bialgebra. He uses this to show for some weak Hopf algebras, one can find some new solutions of the quantum Yang-Baxter equation, in particular when the weak Hopf algebra is a finite Clifford monoid algebra. A semigroup is called Clifford if it is regular and all its idempotents lie in its center. Unfortunately, no examples of finite Clifford monoids are given, nor any examples of weak Hopf algebras, which might lead to new solutions of the quantum Yang-Baxter equation.
scientific article; zbMATH DE number 1221656