NDLI: Differentials of Higher Order in Noncommutative Differential Geometry

Content Provider	Springer Nature Link
Author	Coquereaux, R.
Copyright Year	1997
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 noncommutative differential geometry. For an arbitrary algebra A, people already know how to define the differential algebra ΩA of universal differential forms over A. We define Leibniz forms of order n (these are not forms of degree n, i.e. they are not elements of Ω$^{n}$A) as particular elements of what we call the ‘iterated frame algebra’ of order n, F$_{n}$A, which is itself defined as the2$^{n}$ tensor power of the algebra A. We give a system of generators for this iterated frame algebra and identify the A left-module of forms of order n as a particular vector subspace included in the space of universal 1-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.
Starting Page	241
Ending Page	259
Page Count	19
File Format	PDF
ISSN	03779017
Journal	Letters in Mathematical Physics
Volume Number	42
Issue Number	3
e-ISSN	15730530
Language	English
Publisher	Kluwer Academic Publishers
Publisher Date	1997-01-01
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Group Theory and Generalizations Geometry Mathematical and Computational Physics Statistical Physics
Content Type	Text
Resource Type	Article
Subject	Statistical and Nonlinear Physics Mathematical Physics

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