Higher Order Differential Calculus on $S{L}_q(N)$

Abstract

Let $\Gamma$ be an $N^2$-dimensional bicovariant first order differential calculus on a Hopf algebra $S{L}_q(N)$. There are three possibilities to construct a differential Z-graded Hopf algebra ${\Gamma }^{\wedge }$ which contains $\Gamma$ as its first order part. Let $q$ be a transcendental complex number. For $N>2$ these three Z-graded Hopf algebras 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 $4D_{\pm }$ calculi on $S{L}_q(2)$ the universal calculus is strictly larger than the other two calculi. In particular, the bi-invariant 1-form is not closed.

8 pages, LaTeX2e, uses packages bbm, amsmath, mathrsfs. Presented at the 6th Colloquium "QG & IS", Prag, July 97

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