Connes connection

In mathematics, a Connes connection is a noncommutative generalization of a connection in differential geometry. It was introduced by Alain Connes, and was later generalized by Joachim Cuntz and Daniel Quillen.

Definition

Given a right A-module E, a Connes connection on E is a linear map

\nabla :E\to E\otimes _{A}\Omega ^{1}A

that satisfies the Leibniz rule $\nabla _{r}(sa)=\nabla _{r}(s)a+s\otimes da$ .[1]

Connes, Alain (1995). Noncommutative Geometry. Academic Press. ISBN 978-0-08-057175-1.
Connes, Alain (1985). "Non-commutative differential geometry". Publications Mathématiques de l'IHÉS. 62: 41–144. ISSN 1618-1913.
Cuntz, Joachim; Quillen, Daniel (1995). "Algebra Extensions and Nonsingularity". Journal of the American Mathematical Society. 8 (2): 251–289. doi:10.2307/2152819. ISSN 0894-0347.
García-Beltrán, Dennise; a-Beltrán, Dennise; Vallejo, José A.; Vorobjev, Yuriĭ (2012). "On Lie Algebroids and Poisson Algebras". Symmetry, Integrability and Geometry: Methods and Applications. 8: 006. arXiv:1106.1512. Bibcode:2012SIGMA...8..006G. doi:10.3842/SIGMA.2012.006. S2CID 5946411.
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"Connections". Topics in Cyclic Theory. 2020. pp. 201–228. doi:10.1017/9781108855846.009. ISBN 9781108855846.