arXiv physics/96120151996-12-30math-phhep-thmath.MPmath.QAq-alg

非交换微分几何中的高阶微分

Differentials of higher order in non commutative differential geometry

Robert Coquereaux

备注：: 17 pages, LaTeX

摘要：

在微分几何中，自外微积分诞生以来，符号d^n f以及相应的形式主义就被废弃了。然而，高阶微分是有用的对象，可以用迭代切丛上的函数（或喷流）来解释。我们将这一概念推广到非交换微分几何的情形。对于任意结合代数A，我们已经知道如何定义A上的泛微分形式的微分代数Omega（A）。我们将n阶莱布尼茨形式（这些形式不是n次形式，即它们不是Ω ^n A的元素）定义为n阶“迭代框架代数”F_n A的特殊元素，F_n A本身被定义为代数A的2^n张量幂。我们给出了这个迭代框架代数的生成元系统，并将n阶形式的A-模确定为包含在n-1阶迭代框架代数上的泛1-形式空间中的一个特殊向量子空间。我们研究这些对象的代数结构，恢复的情况下，交换微分阶n（莱布尼茨微分），并给出了几个例子。

英文摘要：

In differential geometry, the notation d^n f along with the corresponding formalism has fallen into disuse since the birth of exterior calculus. However, differentials of higher order are useful objects that can be interpreted in terms of functions on iterated tangent bundles (or in terms of jets). We generalize this notion to the case of non commutative differential geometry. For an arbitrary associative algebra A, one already knows how to define the differential algebra Omega(A) of universal differential forms over A. We define Leibniz forms of order n (these are not forms of degree n, ie they are not elements of Omega^n A) as particular elements of what we call the ``iterated frame algebra'' of order n, F_n A, which is itself defined as the 2^n tensor power of the algebra A. We give a system of generators for this iterated frame algebra and identify the A-module of forms of order n as a particular vector subspace included in the space of universal one-forms built over the iterated frame algebra of order n-1. We study the algebraic structure of these objects, recover the case of the commutative differential calculus of order n (Leibniz differentials) and give a few examples.

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