Convex PBW-type Lyndon Bases And Restricted Two-Parameter Quantum Group Of Type F₄

The article gives the explicit description of PBW type Lyndon bases of two-parameter q-uantum group of type F4, from which some important basic commutation relations of quan-tum root vectors are computed, and then gives an unified property of all commutation rela-tions. At the second chapter, the article mainly discusses the restricted two-parameter quan-tum group when r, s are respectively d-th and d'-th root of unity,l is the least common multi-ple of d and d', we find some of central elements of Ur.s(F4) which are used to construct the re-stricted two-parameter quantum group ur.s(F4). The finite dimensional algebra ur.s(F4) is turn-ed out to be a pointed Hopf algebra. At the third chapter, we find the left and right inte-grals (?) and (?) which play an important role in theory of knot invariants. we get a non-semisimple pointed Hopf subalgebra b of ur.s(F4) byε((?))=ε((?))=0., At last, we prove that ur.s(F4) is a Ribbon Hopf algebra by its Drinfel'd double structure and the results proved by K-auffman and Radford, and as a consequence, we can construct a new kind of quantum invari-ants.