Convex PBW-type Lyndon Bases and Restricted Two-parameter Quantum Group of Type F_(4)

The convex PBW-type Lyndon basis for two-parameter quantum group U_(r,s)(F_(4))is given.Assume thatrs^(-1)is a primitive l-th root of unity with l odd,then the restricted quantum group u_(r,s)(F_(4))as a quotient of U_(r,s)(F_(4))is pointed,and of a Drinfel’d double structure under a certain condition.All of Hopf isomorphisms of u_(r,s)(F_(4))are determined,and the necessary and sufficient condition for u_(r,s)(F_(4))to be a ribbon Hopf algebra is singled out by describing the left and right integrals.