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 (finitedimensional) 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. MSC (2010): 53C15, 15A69 (primary), 57R35, 57R45 (secondary).1 That would be myself, for instance.Given such π : V → X, we endow it with a smooth symmetric bilinear form g : X → V * ⊗ V * (as we have already commented, in general it cannot be a metric, meaning that it does not give a scalar product on individual fibres; we recall that it may not always have the maximal rank possible for a given fibre). We then turn to the subject of our main, which is pseudo-bundles of the corresponding Clifford algebras Cl(V, g) = ∪ x∈X Cl(V x , g(x)) → X and those of (abstract) Clifford modules, where we concentrate on their behavior under the so-called gluing operation. Notice that all our vector spaces (in particular, the fibres π −1 (x) ⊆ V ) are over real numbers.The structure of the paper In order to make the paper self-contained, we collect in Section 1 all the main definitions and facts that are used therein, that is, diffeological spaces, diffeological vector spaces, Clifford algebras and modules, and diffeological algebras. In Section 2, which has somewhat expository nature (this material shall be at least implicitly known to anyone familiar with the diffeology field), we consider some instances of smooth actions of diffeological algebras on diffeological spaces. We comment on diffeological version of Clifford algebras and modules in Section 3. After recalling, in Section 4, the needed facts/constructions regarding diffeological vector pseudo-bundles and pseudo-metrics on them (this material is not new and comes from previous sources), in Sections 5 and 6 we consider the gluing operation for pseudo-bundles of Clifford algebras (Section 5) and those of Clifford modules (Section 6); the main result thus obtained is that, under appropriate compatibility conditions, the result of gluing is again a pseudo-bundle of Clifford algebras/modules in a natural way.
Main definitio...