Classification of bicovariant differential calculi on quantum groups of type A, B, C and D

Under the assumptions that q is not a root of unity and that the differentials duij of the matrix entries span the left module of first order forms, we classify bicovariant differential calculi on quantum groups An-1, Bn, Cn and Dn. 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 An-1 = SLq(n) for n ≥ 3. All these calculi have the dimension n2. For the quantum groups Bn, Cn and Dn we show that except for finitely many q there exist precisely two N2-dimensional bicovariant calculi for N ≥ 3, where N = 2n + 1 for Bn and N = 2n for Cn, Dn. The structure of these calculi is explicitly described and the corresponding ad-invariant right ideals of ker ε are determined. In the limit q → 1 two of the 2n calculi for An-1 and one of the two calculi for Bn, Cn and Dn contain the ordinary classical differential calculus on the corresponding Lie group as a quotient. (orig.)