Diffeological Clifford algebras and pseudo-bundles of Clifford modules

Abstract

We consider the diffeological version of the Clifford algebra of a diffeological finite dimensional vector space; we start by commenting on the notion of a diffeological algebra (which is the expected analogue of the usual one) and that of a diffeological module (also an expected counterpart of the usual notion). After considering the natural diffeology of the Clifford algebra, and considering which of its standard properties re-appear in the diffeological context (most of them), we turn to our main interest, which is constructing the pseudo-bundles of Clifford algebras associated to a given (finite dimensional) diffeological vector pseudo-bundle, and those of the usual Clifford modules (the exterior algebras). The substantial difference that emerges with respect to the standard context, and paves the way to various questions that do not have standard analogues, stems from the fact that the notion of a diffeological pseudo-bundle is very different from the usual bundle, and this under two main respects: it may have fibres of different dimensions, and even if it does not, its total and base spaces frequently are not smooth, or even topological, manifolds.

Keywords:

• Diffeology
• diffeological vector space
• diffeological Clifford algebra
• diffeological dual

Acknowledgements

The creation of this work is due to the influence of a non-mathematician colleague of mine, Professor Riccardo Zucchi, whose good-naturedness, and the ability to provide subtle yet eloquent and inspiring comments and pointers, are beyond any praise. I also have a long-standing debt of gratitude to Prof. Paolo Piazza, to whom I owe my first true encounter with the Atiyah-Singer index theory, and in particular the first real encounter with Clifford algebras. Although the referee reports on the next-to-last version of this paper were OH-polarized (for those who remember some chemistry), I am grateful to both of the authors of those reports. Finally, I am much grateful to the referees of the reports that came in after those, for many useful suggestions.

Notes

No potential conflict of interest was reported by the author.

1 That would be myself, for instance.

2 We will mostly use the latter term, to avoid confusion with the vector space direct sum diffeology (see below), which is a product diffeology defined in the next sentence.

3 This is easy to see directly; assume that a given finite dimensional fine space V is already identified, as a vector space, with an appropriate ${\mathbb{R}}^n$. It suffices to see that the standard diffeology is the finest one on ${\mathbb{R}}^n$. Indeed, any diffeology contains all constant maps, so for any domain $U\subset {\mathbb{R}}^m$ (for whatever m) any diffeology contains the maps $U\ni u\mapsto e_i$, where ${\{e_i\}}_{i=1}^n$ is the canonical basis. Furthermore, any vector space diffeology contains all finite linear combinations with smooth functional coefficients of any collection of its plots. It follows that any usual smooth map $f:U\to {\mathbb{R}}^n$ is a plot for any vector space diffeology on ${\mathbb{R}}^n$, since we can write it as $u\mapsto {\sum }_{i=1}^n{f}_i(u)e_i$. Thus, the standard diffeology is indeed the finest vector space diffeology.

4 Note that the natural embedding of each summand into a direct sum of vector spaces being smooth is a property of the product diffeology coming from vector space diffeologies. There is no analogue of it for a general product diffeology; then again, this is more because for a generic direct product there are no canonical inclusions of the factors.

5 Admittedly, I would not know what to do with a more general case.

6 Recall that a map $h:X\to Y$ between two diffeological spaces is called an induction if it is injective, and the subset diffeology on $h(X)\subseteq Y$ coincides with the pushforward of the diffeology of X by the map h.

7 This mainly means the commutativity between considering the subset diffeology on each fibre and performing a given operation on the individual fibre or, respectively, the entire pseudo-bundle.

8 In the sense of the two points in the base that come into play here.

9 This is the only place where we use the fact that c acts between total spaces of pseudo-bundles.