Differentials of higher order in non commutative differential geometry

Abstract

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.