Convex PBW-type Lyndon bases and restricted two-parameter quantum groups of type \(B\) (Q2268972)

In [Pac. J. Math. 241, No. 2, 243--273 (2009; Zbl 1216.17012)] the first author and \textit{X.-L. Wang} determined all the Hopf algebra isomorphisms of \(\mathfrak u_{r,s} (G_2)\) by describing its sets of left (right) skew-primitive elements. In the paper under review the authors now solve the restricted type B case. In order to study the properties of restriced quantum groups as finite-dimensional Hopf algebras, it is necessary to find nice bases for the corresponding nonrestricted quantum groups at generic cases. Therefore the authors present a direct construction for a convex PBW-type Lyndon basis of type B (for arbitrary rank) and provide explicit information on commutation relations among basis elements that is decisive to single out the left (or right) integrals of \(\mathfrak u_{r,s}(\mathfrak{so}_{2n+1})\). They show that these Hopf algebras are pointed, and determine all Hopf isomorphisms of \(\mathfrak u_{r,s}(\mathfrak{so}_{2n+1})\), as well as \(\mathfrak u_{r,s}(\mathfrak{sl}_n)\) in terms of the set of (left) right skew-primitive elements. Then they prove that \(\mathfrak u_{r,s}(\mathfrak{so}_{2n+1})\) is of Drinfel'd double under a certain condition. Finally, necessary and sufficient conditions for \(\mathfrak u_{r,s}(\mathfrak{sl}_n)\) to be a ribbon Hopf algebra are singled out by describing the left and right integrals.