The Casimir Invariants And Algebraic Structure Of The Centre Of The Quantum Group

Let Uq(g)be the standard Drinfield-Jimbo quantum group over C(q)associated with any simple Lie algebra g.Denote by Uq(g)an extended version of Uq(g)which includes Cartan type elements associated with the integral weights.We describe the structure of the centres of Uq(g)and Uq(g),and construct explicit generators of the centres.Given any finite dimensional Uq(g)-module V of type-1,we construct an infinite family(?)V(k)(k=1,2,…,n)of central elements of Uq(g)following the method of[1,2].The only input into the construction is the universal R-matrix of Uq(g),which is known explicitly,thus the(?)V(k)can be explicitly expressed in terms of elements of Uq(g).One of our new results(see Theorem 3.21)states that the centre of Uq(g)is a polynomial algebra in the generators(?)?i(1)=(?)L(?i)(1)with i=1,2,…,n,where n is the rank of g,the ?i are the fundamental weights,and L(?i)the simple Uq(g)-modules with the highest weights ?i.The proof of the result makes essential use of the quantum Harish-Chandra isomorphism for the quantum group.For g=Cn(n? 2)and Dn(n?3),we take the natural module V=L(?1)for Uq(g),and denote Cn,j the j-th order Casimir element(see(5.39),(5.141)for details).Another new result(see Theorem 5.7 and Theorem 5.13)states that?the center of Uq(sp2n)is generated by the elements Cn,1,Cn,2,…,Cn,n(which are algebraically independent);and?the center of Uq(so2n)is generated by the elements Cn,1,Cn,2,…,Cn,n-2 and(?)L(?n-1)(1),(?)L(?n)(1)(which are algebraically independent).The proof of this result involves several steps.We first determine the eigenvalues of the Cn,j in any irreducible highest weight module(which is a new result in itself),then deduce from the eigenvalues the images Cn,j0 of the elements Cn,j under the quantum Harish-Chandra isomorphism,and finally by using Weyl's character formula,we cast the Cn,j0 in a particularly neat form which enables us to prove the result.Given some specific finite dimensional Uq(g)-module V of type-1,we can also construct an infinite family(?)V(k)(k=1,2,…,n)of central elements of Uq(g)following the method of[1,2].The final new result is Theorem 4.16 on the centre of Uq(g),which shows that?for g=A1,Bn(n? 2),Cn(n? 3),E7,E8,F4 and G2,the center of Uq(g)is a polynomial algebra of n=rank(g)variables(?)L(?,j)(1)for j=1,2,…,n;and?for g=An(n ? 2),D2k+1(k ? 2)and E6,the center of Uq(g)is generated by(?)T(?)(k)for modules T(?)with ? belonging to a known finite set Hilb(M+).The relations among the generators are obtained.