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Connections on central bimodules in noncommutative differential geometry. (English) Zbl 0867.53023

The theory of connections (i.e., of derivation laws) given on commutative \(A\)-modules is extended to noncommutative \(A\)-modules, \(A\) being an associative algebra over \(K=\mathbb{R}\) or \(\mathbb{C}\) with a unit \(1\). The authors define the notion of derivation-based connection on central bimodules (introduced by the authors in [C. R. Acad. Sci., Paris, Sér. I 319, No. 9, 927-931 (1994; Zbl 0829.16028)]). They construct new connections (e.g. tensor product of connections) on central bimodules from given connections on bimodules, and also the gauge transform of a connection \(\nabla\) by a gauge transformation \(g\) on the central bimodule \(M\).
A connection \(\nabla\) on \(\Omega^I_{\text{Der}}(A)\) (where \(\Omega_{\text{Der}}(A)\) is a maximal natural generalization of the graded differential algebra of differential forms which uses \(\text{Der}(A)\) as generalization of vector fields, and \(I\) is some set), will be called linear connection. Its torsion is defined using results from the authors’ papers [ESI Preprint 133 (1994) and LPTHE-ORSAY 94/50 (preprint) (1994)]. After three examples, the authors introduce and study duality between bimodules and modules over the center and apply duality to \(\Omega^I_{\text{Der}}(A)\), \(\text{Der}(A)\) and \(\Omega^I_{\text{Der}}(A)\). Then the reality condition for the case of \(*\)-algebras is studied and finally a noncommutative generalization of pseudo-Riemannian structures is investigated.

MSC:

53C05 Connections (general theory)
46L85 Noncommutative topology
46L87 Noncommutative differential geometry
16W25 Derivations, actions of Lie algebras
53B15 Other connections
16S32 Rings of differential operators (associative algebraic aspects)
16D20 Bimodules in associative algebras

Citations:

Zbl 0829.16028

Software:

KORALZ; TAUOLA
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References:

[1] Cartan, H., Notion d’algèbre différentielle; application aux groupes de Lie et aux variétés où opère un groupe de Lie and La transgression dans un groupe de Lie et dans un espace fibré principal, (Colloque de topologie. Colloque de topologie, Bruxelles, 1950 (1951), Masson: Masson Paris) · Zbl 0045.30601
[2] Connes, A., Non-commutative differential geometry, Publ. IHES, 62, 257 (1986)
[3] Connes, A., Non-commutative Geometry (1994), Academic Press: Academic Press New York · Zbl 0933.46069
[4] Dubois-Violette, M., Dérivation et calcul différentiel non commutatif, C.R. Acad. Sci. Paris Sér. I Math., 307, 403-408 (1988) · Zbl 0661.17012
[5] Dubois-Violette, M., Non-commutative differential geometry, quantum mechanics and gauge theory, (Bartocci, C.; Bruzzo, U.; Cianci, R., Differential Geometric Methods in Theoretical Physics, Proc. Rapallo. Differential Geometric Methods in Theoretical Physics, Proc. Rapallo, 1990. Differential Geometric Methods in Theoretical Physics, Proc. Rapallo. Differential Geometric Methods in Theoretical Physics, Proc. Rapallo, 1990, Lecture Notes in Physics, Vol. 375 (1991), Springer: Springer Berlin) · Zbl 0744.53042
[6] Dubois-Violette, M.; Kerner, R.; Madore, J., Non-commutative differential geometry of matrix algebras, J. Math. Phys., 31, 316 (1990) · Zbl 0704.53081
[7] Dubois-Violette, M.; Kerner, R.; Madore, J., Classical bosons in a non-commutative geometry, Classical Quantum Gravity, 6, 1709 (1989) · Zbl 0675.53071
[8] Dubois-Violette, M.; Madore, J.; Masson, T.; Mourad, J., Linear connections on the quantum plane, Lett. Math. Phys., 35, 351-358 (1995) · Zbl 0835.58003
[9] Dubois-Violette, M.; Michor, P. W., Dérivation et calcul différentiel non commutatif II, C. R. Acad. Sci. Paris Sér. I Math., 319, 927-931 (1994) · Zbl 0829.16028
[10] Dubois-Violette, M.; Michor, P. W., The Frölicher-Nijenhuis bracket for derivation based non commutative differential forms, E.S.I. preprint 113 (1994), L.P.T.H.E.-ORSAY 94/41
[11] Karoubi, M., Homologie cyclique des groupes et algèbres, C. R. Acad. Sci. Sér. I Math., 297, 381-384 (1983) · Zbl 0528.18008
[12] Koszul, J. L., Fibre bundles and differential geometry (1960), Tata Institute of Fundamental Research: Tata Institute of Fundamental Research Bombay · Zbl 0244.53026
[13] Madore, J.; Masson, T.; Mourad, J., Linear connections on matrix geometries, Classical and Quantam Gravity, 12, 1429-1440 (1995) · Zbl 0824.58008
[14] Mourad, J., Linear connections in noncommutative geometry, Classical Quantam Gravity, 12, 965-974 (1995) · Zbl 0822.58006
[15] Rieffel, M., Projective modules over higher dimensional noncommutative tori, Can. J. Math. XL, 2, 257-338 (1988) · Zbl 0663.46073
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