Convex PBW-type Lyndon bases and restricted two-parameter quantum group of type \(F_4\) (Q6113505)

In this paper the authors obtain an explicit description of a convex PBW type Lyndon basis of the two-parameter quantum group \(U_{r, s}(F_4)\) of type \(F_4\); an inductive definition of quantum root vectors of type \(F_4\) is given, using \textit{B. Leclerc}'s co-standard factorization of good Lyndon words as in [Math. Z. 246, No. 4, 691--732 (2004; Zbl 1052.17008)]. A unified description of commutation relations among quantum root vectors imply the existence of certain central elements in the case when parameters \(r, s\) are roots of unity, which generate a Hopf ideal; the corresponding quotient Hopf algebra is called the restricted two-parameter quantum group \(\mathfrak{u}_{r,s}(F_4)\). Properties of these Hopf algebras are studied and in particular they are shown to be pointed. It is shown that \(\mathfrak{u}_{r,s}(F_4)\) is a Drinfel'd double under a certain condition. A necessary and sufficient condition for \(\mathfrak{u}_{r,s}(F_4)\) to be ribbon is given.