On the antipode of a quasitriangular Hopf algebra
Let \((A,R)\) be a quasi-triangular Hopf algebra with antipode \(s\) over a field, where \(R=\sum R^{(1)}\otimes R^{(2)}\) in \(A\otimes A\). Let \(u=\sum s(R^{(2)})R^{(1)}\). The author shows that \(u\) is invertible and \(s^ 2\) is conjugation by \(u\), i.e., \(s^ 2(a)=uau^{-1}\) for \(a\) in \(A\). This implies that \(s\) is bijective. The conjugacy result had been established by \textit{V. G. Drinfel'd} [Algebra Anal. 1, No. 2, 30-46 (1989; Zbl 0718.16035)] assuming that \(s\) was bijective. If \((A,R)\) is also finite-dimensional, the author shows that \(s^ 4\) is conjugation by a group-like element \(h\) of \(A\). In fact, \(h=uv\), where \(v=s(u)^{-1}\). This result had also been proved by Drinfel'd, using different methods. The author's methods are based on his previous work on integrals and trace functions.
scientific article; zbMATH DE number 130645