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

Keywords:

control bimodules; connections

Zbl 0829.16028

KORALZ; TAUOLA
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References:

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