1Theory Group, Nuclear Physics Laboratory, Kharkov National University, Kharkov 61077, Ukraine2Department of Mathematics, Zhejiang University (Xixi Campus) Hangzhou, Zhejiang 310028, China
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Generalization of Hopf algebra Symbol (2) by weakening the invertibility of the generator K, i.e., exchanging its invertibility KK−1 = 1 to the regularity KK = K is studied. Two weak Hopf algebras are introduced: a weak Hopf algebra wSymbol and a J-weak Hopf algebra vSymbol (2) which are investigated in detail. The monoids of group-like elements of wSymbol and vSymbol (2) are regular monoids, which supports the general conjucture on the connection betweek weak Hopf algebras and regular monoids. A quasi-braided weak Hopf algebra Ūwq is constructed from wSymbol. It is shown that the corresponding quasi-R-matrix is regular RwwRwRw = Rw.