半经典微分结构
Semi-classical differential structures
摘要：
我们半经典化的标准概念微分微分学交换几何代数和量子群。在辛情形下，我们证明了微分学的无穷小数据是一个辛联络，并将其曲率解释为外代数的最低阶非结合性。非对易环面的半经典化提供了一个曲率为零的例子。在Poisson-李群的情形下，我们研究了左协变无穷小数据的部分定义的预连接。本文证明了拟三角Poisson-李群的双协变无穷小数据的模空间存在一个典范参考点，该参考点在三角情形下是平坦的。利用Kostant的一个定理，我们完全确定了李代数为单时的模空间：当模空间为一维时，标准预连接是sl_n，n>2以外的唯一点.我们与典型的预连接Drinfeld扭曲，从而将其量子化到一个超coquasi-Hopf外代数。我们还讨论了与Fedosov量化的联系。
英文摘要：
We semiclassicalise the standard notion of differential calculus in noncommutative geometry on algebras and quantum groups. We show in the symplectic case that the infinitesimal data for a differential calculus is a symplectic connection, and interpret its curvature as lowest order nonassociativity of the exterior algebra. Semiclassicalisation of the noncommutative torus provides an example with zero curvature. In the Poisson-Lie group case we study left-covariant infinitesimal data in terms of partially defined preconnections. We show that the moduli space of bicovariant infinitesimal data for quasitriangular Poisson-Lie groups has a canonical reference point which is flat in the triangular case. Using a theorem of Kostant, we completely determine the moduli space when the Lie algebra is simple: the canonical preconnection is the unique point for other than sl_n, n>2, when the moduli space is 1-dimensional. We relate the canonical preconnection to Drinfeld twists and thereby quantise it to a super coquasi-Hopf exterior algebra. We also discuss links with Fedosov quantisation.
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