Abstract
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, which 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 natural map, which in addition turns out to be an isometry for the natural pseudo-metrics on the pseudo-bundles involved.
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References
Chen, K.T.: Iterated path integrals. Bull. Am. Math. Soc. 5, 831–879 (1977)
Christensen, J.D., Sinnamon, G., Wu, E.: The D-topology for diffeological spaces. Pac. J. Math. 272(1), 87–110 (2014)
Christensen, J.D., Wu, E.: Tangent spaces and tangent bundles for diffeological spaces. arXiv:1411.5425v1
Iglesias-Zemmour, P.: Fibrations difféologiques et homotopie. Thèse de doctorat d’État, Université de Provence, Marseille (1985)
Iglesias-Zemmour, P.: Diffeology. Mathematical, Surveys and Monographs, vol. 185. AMS, Providence (2013)
Pervova, E.: Multilinear algebra in the context of diffeology. arXiv:1504.08186v2
Pervova, E.: Diffeological Clifford algebras and pseudo-bundles of Clifford modules. arXiv:1505.06894v2
Pervova, E.: On the notion of scalar product for finite-dimensional diffeological vector spaces. arXiv:1507.03787v1
Pervova, E.: Diffeological vector pseudo-bundles. Topol. Appl. 202, 269–300 (2016)
Pervova, E.: Diffeological gluing of vector pseudo-bundles and pseudo-metrics on them. Topol. Appl. 220, 65–99 (2017)
Pervova, E.: Diffeological Dirac operators and diffeological gluing. arXiv:1701.06785v1
Satake, I.: The Gauss-Bonnet theorem for V-manifolds. J. Math. Soc. Jpn. 9(4), 464–492 (1957)
Souriau, J.M.: Groups différentiels. Differential geometrical methods in mathematical physics (Proc. Conf., Aix-en-Provence/Salamanca, 1979). Lecture Notes in Mathematics, vol. 836, pp. 91–128. Springer (1980)
Souriau, J.M.: Groups différentiels de physique mathématique. South Rhone seminar on geometry, II (Lyon, 1984), Astérisque 1985, Numéro Hors Série, pp. 341–399
Vincent, M.: Diffeological differential geometry. Master Thesis, University of Copenhagen (2008)
Watts, J.: Diffeologies, differential spaces, and symplectic geometry. PhD Thesis, 2012, University of Toronto, Canada
Wu, E.: Homological algebra for diffeological vector spaces. Homol. Homotopy Appl. 17(1), 339–376 (2015)
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Communicated by Rafał Abłamowicz
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Pervova, E. Pseudo-bundles of Exterior Algebras as Diffeological Clifford Modules. Adv. Appl. Clifford Algebras 27, 2677–2737 (2017). https://doi.org/10.1007/s00006-017-0769-z
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DOI: https://doi.org/10.1007/s00006-017-0769-z