Restricted Two-Parameter Quantum Groups Of Type C

In the paper,we study the two-parameter quantum groups U_r,_s(sp_2n)discovered by Bergeron-Gao-Hu[14],construct a convex PBW-type basis and do a crucial and hard work to give a detailed description of the commutative relations between these basis elements.Then,we restrict the parameter r and s of two-parameter quantum groups U_r._s(sP_2n)to be roots of unity,and construct a family of finite-dimensional Hopf algebras U_r,_s(sp_2n)of dimension l²ⁿ²⁺²ⁿas a quotient of U_r,_s(sp_2n)by a Hopf ideal I_n,which is generated by the central elements.Next,we show that these Hopfalgebras are pointed and of Drinfel'd doubles of a cer-tain Hopf subalgebra b.Our results on the skew-primitive elements of U_r,_s(sP_2n)enable us to determine when two such restricted two-parameter quantum groups are isomorphic. We determine the left and right integrals of b and use these results in combining with a result of L.H.Kauffmann and D.E.Radford from[53]to give necessary and suffi-cient conditions for these Hopf algebras to possess a ribbon element.Indeed,the ribbon element of these pointed Hopf algebras provides a new quantum invariant of knots and links.Since the classification of the finite-dimensional Hopf algebras seems very far off, it is quite useful to have various means of constructing examples of finite-dimensional Hopf algebras.Our results on restricted two-parameter quantum groups of type C provide many new examples of finite-dimensional Hopf algebras.