On the notion of scalar product for finite-dimensional diffeological vector spaces

It is known that the only finite-dimensional diffeological vector space that admits a diffeologically smooth scalar product is the standard space of appropriate dimension. In this note we consider a way to circumnavigate this issue, by introducing a notion of pseudo-metric, which, said informally, is the least-degenerate symmetric bilinear form on a given space. We apply this notion to make some observation on subspaces which split off as smooth direct summands (providing examples which illustrate that not all subspaces do), and then to show that the diffeological dual of a finite-dimensional diffeological vector space always has the standard diffeology and in particular, any pseudo-metric on the initial space induces, in the obvious way, a smooth scalar product on the dual.Comment: 9 pages; various improvements throughout, several proof modified at a referee's suggestio

Groups of tree automorphisms as diffeological groups

We consider certain groups of tree automorphisms as so-called diffeological groups. The notion of diffeology, due to Souriau, allows to endow non-manifold topological spaces, such as regular trees that we look at, with a kind of a differentiable structure that in many ways is close to that of a smooth manifold; a suitable notion of a diffeological group follows. We first study the question of what kind of a diffeological structure is the most natural to put on a regular tree in a way that the underlying topology be the standard one of the tree. We then proceed to consider the group of all automorphisms of the tree as a diffeological space, with respect to the functional diffeology, showing that this diffeology is actually the discrete one, the fact that therefore is true for its subgroups as well.Comment: 11 pages, 1 figure; rather minor changes with respect to the previous versio

Diffeological Clifford algebras and pseudo-bundles of Clifford modules

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 its expected properties, we turn to our main interest, which is constructing pseudo-bundles of diffeological Clifford algebras and those of diffeological Clifford modules, by means of the procedure called diffeological gluing. The paper has a significant expository portion, regarding mostly diffeological algebras and diffeological vector pseudo-bundles.Comment: 35 pages; exposition improved, an example adde

Differential 1-forms on diffeological spaces and diffeological gluing

This paper aims to describe the behavior of diffeological differential 1-forms under the operation of gluing of diffeological spaces along a smooth map. In the diffeological context, two constructions regarding diffeological forms are available, that of the vector space \Omega^1(X) of all 1-forms, and that of the (pseudo-)bundle \Lambda^1(X) of values of 1-forms. We describe the behavior of the former under an arbitrary gluing of two diffeological spaces, while for the latter, we limit ourselves to the case of gluing along a diffeomorphism.Comment: 31 pages; except for Sections 1-3, the paper has entirely reworked. There was an important omission in the statements of the main results; this is now fixed. I am still not sure to have included all the correct references. If your work (or one you know of) is not cited here and it should be, it's because I am not aware of it, so please let me kno

Diffeological vector pseudo-bundles

We consider a diffeological counterpart of the notion of a vector bundle (we call this counterpart a pseudo-bundle, although in the other works it is called differently; among the existing terms there are a "regular vector bundle" of Vincent and "diffeological vector space over X" of Christensen-Wu). The main difference of the diffeological version is that (for reasons stemming from the independent appearance of this concept elsewhere), diffeological vector pseudo-bundles may easily not be locally trivial (and we provide various examples of such, including those where the underlying topological bundle is even trivial). Since this precludes using local trivializations to carry out many typical constructions done with vector bundles (but not the existence of constructions themselves), we consider the notion of diffeological gluing of pseudo-bundles, which, albeit with various limitations that we indicate, provides when applicable a substitute for said local trivializations. We quickly discuss the interactions between the operation of gluing and typical operations on vector bundles (direct sum, tensor product, taking duals) and then consider the notion of a pseudo-metric on a diffeological vector pseudo-bundle.Comment: 29 pages, no figure

Pseudo-bundles of exterior algebras as diffeological Clifford modules

We consider the diffeological pseudo-bundles of exterior algebras, and the Clifford action of the corresponding Clifford algebras, associated to a given finite-dimensional and locally trivial diffeological vector pseudo-bundle, as well as the behavior of the former three constructions (exterior algebra, Clifford action, Clifford algebra) under the diffeological gluing of pseudo-bundles. Despite these being our main object of interest, we dedicate significant attention to the issues of compatibility of pseudo-metrics, and the gluing-dual commutativity condition, that is, the condition ensuring that the dual of the result of gluing together two pseudo-bundles can equivalently be obtained by gluing together their duals (this is not automatic in the diffeological context). We show that, assuming that the dual of the gluing map, which itself does not have to be a diffeomorphism, on the total space is one, the commutativity condition is satisfied, via a very natural map, which in addition turns out to be an isometry for the natural pseudo-metrics on the pseudo-bundles involved.Comment: 48 pages; various minor improvements throughout, references update

Linear and multilinear algebra on diffeological vector spaces

This paper deals with some basic constructions of linear and multilinear algebra on finite-dimensional diffeological vector spaces. We consider the diffeological dual formally checking that the assignment to each space of its dual defines a covariant functor from the category of finite-dimensional diffeological vector spaces to the category of standard (that is, carrying the usual smooth structure) vector spaces. We verify that the diffeological tensor product enjoys the typical properties of the usual tensor product, after which we focus on the so-called smooth direct sum decompositions, a phenomenon exclusive to the diffeological setting. We then consider the so-called pseudo-metrics (diffeological analogues of scalar products), and discuss the consequent decompositions of vector spaces into a smooth direct sum of the maximal isotropic subspace and a characteristic subspace; we show that such a decomposition actually depends only on the choice of the coordinate system. Furthermore, we show that, contrary to what was erroneously claimed (by me) elsewhere, a characteristic subspace is not unique and is not invariant under the diffeomorphisms of the space on itself. The maximal isotropic subspace is on the other hand an invariant of the space itself, and this fact allows to assign to each its well-defined characteristic quotient, obtaining another functor, this time a contravariant one, to the category of standard vector spaces. After discussing the diffeological analogues of isometries, we end with some remarks concerning diffeological algebras and diffeological Clifford algebras. Perhaps a larger than usual part of the paper recalls statements that already appear elsewhere, but when this is the case, we try to accompany them with new proofs and examples.Comment: 24 pp; almost entirely rewritten, and many new parts adde

Diffeological gluing of vector pseudo-bundles and pseudo-metrics on them

Although our main interest here is developing an appropriate analog, for diffeological vector pseudo-bundles, of a Riemannian metric, a significant portion is dedicated to continued study of the gluing operation for pseudo-bundles introduced in arXiv:1509.03023. We give more details regarding the behavior of this operation with respect to gluing, also providing some details omitted from arXiv:1509.03023, and pay more attention to the relations with the spaces of smooth maps. We also show that a usual smooth vector bundle over a manifold that admits a finite atlas can be seen as a result of a diffeological gluing, and thus deduce that its usual dual bundle is the same as its diffeological dual. We then consider the notion of a pseudo-metric, the fact that it does not always exist (which seems to be related to non-local-triviality condition), construction of an induced pseudo-metric on a pseudo-bundle obtained by gluing, and finally, the relation between the spaces of all pseudo-metrics on the factors of a gluing, and on its result. We conclude by commenting on the induced pseudo-metric on the pseudo-bundle dual to the given one.Comment: 31 pages; title changed, an omission in the proof of Theorem 2.3 corrected, an erroneous claim (that was not used anywhere in the paper) taken out of the statement of Lemma 4.1, Section 3 shortened for expository purposes, various minor improvements throughou