Calculus Revisited pp 85-122 | Cite as

# Differential Calculi

## Abstract

We go now to differential calculi and geometry involving quantum groups and Hopf algebras following [**7, 29, 61, 84, 95, 96, 97, 98, 99, 100, 101, 149, 155, 191, 192, 193, 216, 400, 457, 461, 462, 469, 473, 478, 479, 567, 602, 608, 666] (cf. also [23, 145, 311, 312, 346, 441, 640, 672**]). The subject has a long history, going back to the seminal paper [**666**] at least (on which much is dependent), and is of great importance in developing gauge theory and quantum gravity today (cf. [**457**]).Evidently one needs a first order differential calculus or cotangent bundle before trying to do any geometry or gauge theory. However there is considerable nonuniqueness involved and many possible constructions so that any sort of complete description takes a long time (cf. especially [**96, 100** **101, 400, 457, 666**]).We will try to shortcut this a bit by selecting only a few background items leading to [**457**], which seems to represent where one wants to go in order to generate applications to quantum physics and gravity. We have made several attempts to organize matters starting from [**100**, **101, 473**] but each time we are led back to [**666**] for background information. Hence we will simply begin with the excellent exposition from [**666**] and try to give enough detail so that a genuine insight into quantum differential calculi can be obtained. Some notation is changed to agree with [**100, 101, 473**].

## Keywords

Hopf Algebra Associative Algebra Differential Calculus Exterior Algebra Differential Ideal## Preview

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