Higher order differential calculus on SL _q(N)

Abstract

Let Γ be a bicovariant first order differential calculus on a Hopf algebra $\mathcal{A}$. There are three possibilities to construct a differential N ₀-graded Hopf algebra Γ which contains Γ as its first order part. In all cases Γ ^{∧ ∧} is a quotient Γ ^∧ = Γ ^⊗/J of the tensor algebra by some suitable ideal. We distinguish three possible choices _u J, _s J, and _W J, 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 N ²-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.