Higher order differential calculus on SL q(N)

Let be a bicovariant first order differential calculus on a Hopf algebra . There are three possibilities to construct a differential N₀-graded Hopf algebra which contains as its first order part. In all cases is a quotient = /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 ) and the last one is Woronowicz' external algebra. Let q be a transcendental complex number and let be one of the N²-dimensional bicovariant first order differential calculi on the quantum group SL_q(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 SL_q(2) the universal calculus is strictly larger than the other two calculi. In particular, the bi-invariant 1-form is not closed.