A left quantum group.
A left Hopf algebra is a bialgebra \(H\) with a left antipode \(S\), i.e. a linear map \(S\colon H\to H\) such that \(S*I=\mu\varepsilon\), where \(\varepsilon\) is the counit, \(\mu\) is the unit, \(I\) is the identity map, and \(*\) is the convolution product. A left Hopf algebra which is not a Hopf algebra was constructed by \textit{J. A. Green, W. D. Nichols} and \textit{E. J. Taft} [J. Algebra 65, 399-411 (1980; Zbl 0439.16008)]. In the paper under review, the authors construct a left Hopf algebra which is not a Hopf algebra by considering only some of the defining relations of \(\text{SL}_q(2)\). They obtain a left Hopf algebra \(H\) with the property that no left antipode of \(H\) is an algebra antimorphism.
scientific article; zbMATH DE number 2159741