Higher order differential calculus on SLq(N)

Let Γ be a bicovariant first order differential calculus on a Hopf algebra A. There are three possibilities to construct a differential N0-graded Hopf algebra Γcirconflex which contains Γ as its first order part. In all cases Γcirconflex is a quotient Γcirconflex = Γx/J of the tensor algebra by some suitable ideal. We distinguish three possible choices uJ, sJ, and wJ, where the first one generates the universal differential calculus (over Γ) and the last one is Woronowicz' external algebra. Let q be a transcendental complex number and let Γ be one of the N2-dimensional bicovariant first order differential calculi on the quantum group SLq(N). Then for N ≥ 3 the three ideals coincide. For Woronowicz' external algebra we calculate the dimensions of the spaces of left-invariant and bi-invariant k-forms. In this case each bi-invariant form is closed. In case of 4D± calculi on SLq(2) the universal calculus is strictly larger than the other two calculi. In particular, the bi-invariant 1-form is not closed. (author)