Higher order differential calculus on SL q(N) |
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Authors: | I. Heckenberger A. Schüler |
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Affiliation: | (1) Institute of Mathematics, University of Leipzig, Augustusplatz 10, 04109 Leipzig, Germany |
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Abstract: | 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 ![Gamma](/content/p047723755550533/xxlarge915.gif) ![and](/content/p047723755550533/xxlarge8743.gif) is a quotient ![Gamma](/content/p047723755550533/xxlarge915.gif) = ![Gamma](/content/p047723755550533/xxlarge915.gif) /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 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. |
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