Weak Hopf algebras and singular solutions of quantum Yang-Baxter equation (Q5961017)

A bialgebra \(H\) is called a weak Hopf algebra if it has a weak antipode, i.e., a linear endomorphism \(T\) satisfying \(I*T*I=I\) and \(T*I*T=T,*\) the convolution product. The authors construct a weak Hopf algebra \(wsl_q(2)\) which is a modification of \(sl_q(2)\). The latter has generators \(E,F,K\) and \(K^{-1}\). In \(wsl_q(2)\), \(K^{-1}\) is replaced by \(\overline K\) satisfying \(K\overline KK= K\) and \(\overline KK\overline K=\overline K\). A certain homomorphic image of \(wsl_q(2)\) has a quasi-\(R\)-matrix (the quasi-commutative property is replaced by \(\Delta^{op}(x) R+R\Delta(x))\). The group-like elements of \(wsl_q(2)\) are identified as \(K^i\overline K^j\) for \(i,j\geq 0\), and it is shown that they form a regular monoid. Another variation \(vsl_q(2)\) is presented, focusing on the elements \(J=K\overline K\) and \(\overline J=\overline KK\). \(vsl_q(2)\) is not a weak Hopf algebra. Its group-like elements also form a regular monoid.