On the canonical connection for smooth envelopes

• Autorzy: Moreno G.
• Języki publikacji: EN

Abstrakty

EN

A notion known as smooth envelope, or superposition closure, appears naturally in several approaches to generalized smooth manifolds, which were proposed in the last decades. Such an operation is indispensable in order to perform differential calculus. A derivation of the enveloping algebra can be restricted to the original one, but it is a delicate question if the the vice–versa can be done as well. In a physical language, this would correspond to the existence of a canonical connection. In this paper, we show an example of an algebra which always possesses such a connection.

Słowa kluczowe

EN

smooth envelope; canonical connection; differential calculus; algebra

PL

połączenia kanoniczne; rachunek różniczkowy; algebra

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2014
• Tom: Vol. 47, nr 2
• Strony: 459--464
• Opis fizyczny: Bibliogr. 14 poz.

Twórcy

autor

Moreno G.

• Mathematical Institute in Opava Silesian University in Opava Na Rybnicku 626/1, 746 01 Opava, Czech Republic

Bibliografia

• [1] A. De Paris, A. Vinogradov, Fat Manifolds and Linear Connections, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2009.
• [2] L. Gillman, M. Jerison, Rings of Continuous Functions, Reprint of the 1960 edition; Graduate Texts in Mathematics, No. 43, Springer-Verlag, New York, 1976.
• [3] A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas., Inst. Hautes Études Sci. Publ. Math., 20 (1964).
• [4] M. Heller, Algebraic foundations of the theory of differential spaces, Differential spaces and their applications (Pasierbiec, 1990), Demonstratio Math. 24 (1991), 349–364.
• [5] T. Jagodzinski, W. Sasin, Weil homomorphism in non-commutative differential spaces, Demonstratio Math. 38 (2005), 507–516.
• [6] J. Johnson, Kähler differentials and differential algebra, Ann. of Math. (2) 89 (1969), 92–98.
• [7] J. Krasil’shchik, A. Verbovetsky, Homological methods in equations of mathematical physics, Open Education & Sciences, Opava, 1998. http://arxiv.org/abs/math/9808130v2
• [8] P. W. Michor, Manifolds of Differentiable Mappings, Shiva Mathematics Series, 3, Shiva Publishing Ltd., Nantwich, 1980.
• [9] J. W. Milnor, J. D. Stasheff, Characteristic Classes, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, N. J., 1974.
• [10] J. Nestruev, Smooth Manifolds and Observables, Graduate Texts in Mathematics, 220, Joint work of A. M. Astashov, A. B. Bocharov, S. V. Duzhin, A. B. Sossinsky, A. M. Vinogradov and M. M. Vinogradov; Translated from the 2000 Russian edition by Sossinsky, I. S. Krasil’ schik and Duzhin, Springer-Verlag, New York, 2003.
• [11] G. Sardanashvily, Lectures on Differential Geometry of Modules and Rings, 2009, arXiv:0910.1515
• [12] R. Sikorski, Differential modules, Colloq. Math. 24 (1971/72), 45–79.
• [13] R. G. Swan, Vector bundles and projective modules, Trans. Amer. Math. Soc. 105 (1962), 264–277.
• [14] E. M. Vechtomov, On the general theory of rings of continuous functions, Uspekhi Mat. Nauk 49 (1994), 177–178.