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

Under the assumptions thatq is not a root of unity and that the differentialsdu_jⁱ of the matrix entries span the left module of first order forms, we classify bicovariant differential calculi on quantum groupsA_n–1,B_n,C_n andD_n. We prove that apart one dimensional differential calculi and from finitely many values ofq, there are precisely2n such calculi on the quantum groupA_n–1=SL_q(n) forn3. All these calculi have the dimensionn². For the quantum groupsB_n,C_n andD_n we show that except for finitely manyq there exist precisely twoN²-dimensional bicovariant calculi forN3, whereN=2n+1 forB_n andN=2n forC_n,D_n. The structure of these calculi is explicitly described and the corresponding ad-invariant right ideals of ker are determined. In the limitq1 two of the 2n calculi forA_n–1 and one of the two calculi forB_n,C_n andD_n contain the ordinary classical differential calculus on the corresponding Lie group as a quotient.