Two-parameter Quantum Groups Of Type D With Lyndon Bases And Commutation Relations

In this paper, we investigate the two-parameter quantum groups corresponding to Lie algebras of type D with convex PBW-type Lyndon Bases and commutation relations. To begin with, we define the associate algebra Ur,s(so2n) in Q(r, s) by generators and relations. Moreover, we restrict the parameter r and s to be roots of unity, which is the two-parameter quantum group of type D we studied here, we get the construction of quantum root vectors inductively using the good Lyndon words together with (r, s)-brackets for Ur,s(so2n). Furthermore, we calculate the commutation relations among the Lyndon bases elements εi,j (1≤i≤j≤n-1), en and εi,j’(1≤i(≠n-1)<j≤n), as well as εli,j(1≤i≤j≤n-1), eln and εli,j’(1≤i(≠n-1)<j≤n) in detail, as a result, we obtain some homogeneous central elements of degree l for Ur,s(so2n).