CLASSIFICATION OF BICOVARIANT DIFFERENTIAL CALCULI ON QUANTUM GROUPS OF TYPE-A, TYPE-B, TYPE-C AND TYPE-D

Under the assumptions that q is not a root of unity and that the differentials du(j)(i) of the matrix entries span the left module of first order forms, we classify bicovariant differential calculi on quantum groups A(n-1), B-n, C-n and D-n. We prove that apart one dimensional differential calculi and from finitely many values of q, there are precisely 2n such calculi on the quantum group A(n-1) = SL(q)(n) for n greater than or equal to 3. All these calculi have the dimension n(2). For the quantum groups B-n, C-n and D-n we show that except for finitely many q there exist precisely two N-2-dimensional bicovariant calculi for N greater than or equal to 3, where N = 2n + 1 for B-n and N = 2n for C-n, D-n. The structure of these calculi is explicitly described and the corresponding ad-invariant right ideals of ker epsilon are determined. In the limit q --> 1 two of the 2n calculi for A(n-1) and one of the two calculi for B-n, C-n and D-n contain the ordinary classical differential calculus on the corresponding Lie group as a quotient.