УДК 539.12 REGULAR R-MATRIX AND WEAK HOPF ALGEBRAS Duplij S.A. Kharkov National University, Kharkov 61077, Ukraine Li Fang Department of Mathematics, Zhejiang University (Xixi Campus), Hangzhou, Zhejiang 310028, China Weak Hopf algebras as generalizations of Hopf algebras [1] were introduced in [2], where its characterizations and applications were also studied. A k -bialgebra ( , , , ,) H H μ η ε = Δ with multiplication μ , unity η , counity ε , comultiplication Δ , is called a weak Hopf algebra if there exists Hom ( , ) k T HH ∈ such that * * id T id id = , * * T id T T = where T is called a weak antipode of H . One of its aims is to construct some singular solutions of the quantum Yang-Baxter equation (QYBE) [2]. We study here generalization of Hopf algebra ( ) 2 q sl by weakening the invertibility of the generator K , i.e. exchanging its invertibility 1 1 KK − = to the regularity KKK K = .Here we investigate a weak Hopf algebra () 2 q wsl and a J -weak Hopf algebra ( ) 2 q vsl as generalizations of () 2 q sl and non-trivial examples of weak Hopf algebras [2]. A quasi-braided weak Hopf algebra w q U from () 2 q wsl is constructed whose quasi- R -matrix is regular [3]. Let q C ∈ and 1 q ≠± , 0 . The quantum enveloping algebra ( ) (2) q q q U U sl = (see [6]) is generated by four variables(Chevalley generators) E , F , K , 1 K − with the relations 1 1 1 K K KK − − = = , 1 2 KEK qE − = , 1 2 KFK q F − − = , 1 1 K K EF FE q q − − − − = − . Now we try to weaken the invertibility of K to regularity, as usually in semigroup theory [4] (see also [5] for higher regularity). It can be done in two different ways. Define (2) w q q U wsl = , which is called a weak quantum algebra, as the algebra generated by the four variables w E , w F , w K , w K with the relations: 2 2 2 2 1 , , , , , , , . w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w KK KK KKK K KKK K KE qEK KE q EK KF q FK KF qFK K K EF FE q q − − − = = = = = = = − − = − Define (2) v q q U vsl = , which is called a J -weak quantum algebra, as the algebra generated by the four variables v E , w F , v K , v K with the relations ( v v v J KK = ): 2 2 1 , , , , , . v v v v v v v v v v v v v v v v v v v v v v v v v v v v KK KK KKK K KKK K K K KEK qE KFK q F EJF FJE q q − − = = = − = = − = − Let w w w J KK = . List some useful properties of w J which will be needed below. Firstly, 2 w w J J = , which means that w J is a projector. For any variable X , define “ J -conjugation” as w J w w X J XJ = , and