Convex PBW-type Lyndon Bases and Restricted Two-parameter Quantum Group of Type F4

The convex PBW-type Lyndon basis for two-parameter quantum group Ur,s (F4)is given. Assume that rs−1 is a primitive ℓ-th root of unity with ℓ odd, then the restricted $${\mathfrak{u}_{r,s}}({F_4})$$ as quotient Ur,s(F4) pointed, and Drinfel’d double structure under certain condition. All Hopf isomorphisms are determined, necessary sufficient condition to be ribbon algebra singled out by describing left right integrals.