Weak Hopf Algebras and Singular Solutions¶of Quantum Yang–Baxter Equation

We investigate a generalization of Hopf algebra by weakening the invertibility of the generator K, i.e. exchanging its invertibility KK ^{− 1}= 1 to the regularity . This leads to a weak Hopf algebra and a J-weak Hopf algebra which are studied in detail. It is shown that the monoids of group-like elements of and are regular monoids, which supports the general conjucture on the connection betweek weak Hopf algebras and regular monoids. Moreover, from a quasi-braided weak Hopf algebra is constructed and it is shown that the corresponding quasi-R-matrix is regular . Received: 1 May 2001 / Accepted: 1 September 2001