Differential calculus and connections on a quantum plane at a cubic root of unity

Details Differential calculus and connections on a quantum plane at a cubic root of unity Differential calculus and connections on a quantum plane at a cubic root of unity

1998-07-12

arxiv.org/abs/math-ph/9807012v2

Rev. Math. Phys. 12 (2000), 227-285

Physics

Mathematical Physics

53 pages, 1 small PS figure, Latex, diagrams.sty (included); few errors corrected, final version to appear in Rev. Math. Phys

Scientific paper

We consider the algebra of N x N matrices as a reduced quantum plane on which a finite-dimensional quantum group H acts. This quantum group is a quotient of U_q(sl(2,C)), q being an N-th root of unity. Most of the time we shall take N=3; in that case \dim(H) = 27. We recall the properties of this action and introduce a differential calculus for this algebra: it is a quotient of the Wess-Zumino complex. The quantum group H also acts on the corresponding differential algebra and we study its decomposition in terms of the representation theory of H. We also investigate the properties of connections, in the sense of non commutative geometry, that are taken as 1-forms belonging to this differential algebra. By tensoring this differential calculus with usual forms over space-time, one can construct generalized connections with covariance properties with respect to the usual Lorentz group and with respect to a finite-dimensional quantum group.

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