单位立方根上量子平面上的微分学和联系
Differential calculus and connections on a quantum plane at a cubic root of unity
摘要：
我们把N × N矩阵代数看作一个约化的量子平面，有限维量子群H作用于该平面上.这个量子群是U_q（sl（2，C））的商，q是单位的N次根。大多数情况下，我们取N = 3;在这种情况下，\dim（H）= 27。我们回顾这个行动的性质，并介绍了微分代数：它是一个商的韦斯-祖米诺复杂。量子群H也作用在相应的微分代数上，我们用H的表示论研究了它的分解。我们还调查的性质的连接，在非交换几何的意义上，被视为1-形式属于这个微分代数。通过在时空上用通常的形式张量这个微分，人们可以构造关于通常的洛伦兹群和关于有限维量子群具有协方差性质的广义联系。
英文摘要：
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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