Factorization Of Simple Modules For Restricted Two-parameter Quantum Group Of Type D And Commutation Relations For Two-parameter Quantum Group Of E₆

For the restricted two-parameter quantum group of type D,we give some conditions in which a simple module can be factored as the tensor product of a simple module of the quotient algebra u_r,s(so_2n)by central group-like elements and a one-dimensional simple module of u_r,s(so_2n).Let θ be a primitive l th root of unity,r = θ^p,s = θ^q,so a simple module of u_θp,θq(so²ⁿ)can be factorizable as above if and only if（4（p-q）,l）= 1·In order to study small quantum group of type E₆,We should prove the vectors E_βi^l,F_βi^l（1≤i<36）are central in E₆,where we suppose that r is d-th root of unity,s is d-th of unity,and l is the least common multiple of d and d^’.Then we try to give the commutation relations between the quantum root vectors E_βj（1≤j≤36）and the generators e_i,f_i w_i^±i,w_i^’±i（1≤i≤6）which are necessary to induce the restricted type.