Differential Calculi on Commutative Algebras

TL;DR

This paper systematically investigates differential calculi on commutative algebras, establishing their structure, classifications, and connections to stochastic calculus, with implications for geometric and physical theories.

Contribution

It introduces a correspondence between first order differential calculi and associative products in 1-forms, and classifies calculi with constant structure functions for algebras generated by coordinates.

Findings

Established a correspondence between differential calculi and associative products in 1-forms.

Classified calculi with constant structure functions for n<4.

Analyzed reducibility and extensions of these calculi.

Abstract

A differential calculus on an associative algebra A is an algebraic analogue of the calculus of differential forms on a smooth manifold. It supplies A with a structure on which dynamics and field theory can be formulated to some extent in very much the same way we are used to from the geometrical arena underlying classical physical theories and models. In previous work, certain differential calculi on a commutative algebra exhibited relations with lattice structures, stochastics, and parametrized quantum theories. This motivated the present systematic investigation of differential calculi on commutative and associative algebras. Various results about their structure are obtained. In particular, it is shown that there is a correspondence between first order differential calculi on such an algebra and commutative and associative products in the space of 1-forms. An example of such a product is provided by the Ito calculus of stochastic differentials. For the case where the algebra A is freely generated by `coordinates' x^i, i=1,...,n, we study calculi for which the differentials dx^i constitute a basis of the space of 1-forms (as a left A-module). These may be regarded as `deformations' of the ordinary differential calculus on R^n. For n < 4 a classification of all (orbits under the general linear group of) such calculi with `constant structure functions' is presented. We analyse whether these calculi are reducible (i.e., a skew tensor product of lower-dimensional calculi) or whether they are the extension (as defined in this article) of a one dimension lower calculus. Furthermore, generalizations to arbitrary n are obtained for all these calculi.