Pseudo-bundles of Exterior Algebras as Diffeological Clifford Modules

Pervova, Ekaterina

doi:10.1007/s00006-017-0769-z

Cited by 3 publications

(4 citation statements)

References 17 publications

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“…In fact, this is an instance of the similar commutativity of gluing with the construction of the exterior algebras' pseudo-bundles (see [10]); once again, all formulae are consistent with each other on all intersection subsets.…”

Section: The Exterior Algebras V and V I As Clifford Modulesmentioning

confidence: 67%

“…Thus, another necessary condition for the gluing-dual commutativity is that the dual mapf * be a diffeomorphism of its domain with its image. It is shown in [10] that this condition becomes sufficient under the additional assumption that the factors of gluing admit the so-called compatible pseudo-metrics (see the next section).…”

Section: Gluing and Other Operationsmentioning

confidence: 99%

“…In the theorem below we assume that π 1 : V 1 → X 1 , π 2 : V 2 → X 2 , g 1 , and g 2 are as in Theorem 5.4, but the gluing pair (f , f ) is arbitrary (assuming that it is suitable for gluing, of course). Also notice that the fact of the existence of pseudo-metrics compatible with it does impose significant restrictions onf (which, on the other hand, may still have a non-trivial kernel, see [10]).…”

Section: Non-diffeomorphismfmentioning

confidence: 99%

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Diffeological Clifford algebras and pseudo-bundles of Clifford modules

Pervova

2018

Linear and Multilinear Algebra

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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...

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Section: The Exterior Algebras V and V I As Clifford Modulesmentioning

confidence: 67%

Section: Gluing and Other Operationsmentioning

confidence: 99%

Section: Non-diffeomorphismfmentioning

confidence: 99%

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Diffeological Clifford algebras and pseudo-bundles of Clifford modules

Pervova

2018

Linear and Multilinear Algebra

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“…which is a diffeological vector pseudo-bundle for its standard diffeology based on the tensor product diffeology. Then the standard Clifford action c Λ of Cℓ(Λ 1 (X), g Λ ) on (Λ 1 (X)) is smooth ( [16]); furthermore, the connection ∇ Λ induces (in a completely standard way) a connection ∇ (Λ) on (Λ 1 (X)).…”

Section: The Definition Of the De Rham Operatormentioning

confidence: 99%

Diffeological De Rham operators

Pervova¹

2017

Preprint

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We consider the notion of the De Rham operator on finite-dimensional diffeological spaces such that the diffeological counterpart Λ 1 (X) of the cotangent bundle, the so-called pseudo-bundle of values of differential 1-forms, has bounded dimension. The operator is defined as the composition of the Levi-Civita connection on the exterior algebra pseudo-bundle (Λ 1 (X)) with the standardly defined Clifford action by Λ 1 (X); the latter is therefore assumed to admit a pseudo-metric for which there exists a Levi-Civita connection. Under these assumptions, the definition is fully analogous the standard case, and our main conclusion is that this is the only way to define the De Rham operator on a diffeological space, since we show that there is not a straightforward counterpart of the definition of the De Rham operator as the sum d + d * of the exterior differential with its adjoint. We show along the way that other connected notions do not have full counterparts, in terms of the function they are supposed to fulfill, either; this regards, for instance, volume forms, the Hodge star, and the distinction between the k-th exterior degree of Λ 1 (X) and the pseudo-bundle of differential k-forms Λ k (X). MSC (2010): 53C15 (primary), 57R35 (secondary).

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Smooth constructions of homotopy-coherent actions

Tanaka

2022

Algebr. Geom. Topol.

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Pseudo-bundles of Exterior Algebras as Diffeological Clifford Modules

Cited by 3 publications

References 17 publications

Diffeological Clifford algebras and pseudo-bundles of Clifford modules

Diffeological Clifford algebras and pseudo-bundles of Clifford modules

Diffeological De Rham operators

Smooth constructions of homotopy-coherent actions

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