Indian Journal of Pure and Applied Mathematics

, Volume 50, Issue 4, pp 891–901

# Connections on smooth Lie algebra bundles

• K. Ajaykumar
• B. S. Kiranagi
Article

## Abstract

We define the notion of Lie Ehresmann connection on Lie algebra bundles and show that a Lie connection on a Lie algebra bundle induces a Lie Ehresmann connection. The converse is proved for normed Lie algebra bundles. We then show that the connection on adjoint bundle corresponding to the connection on principal G—bundle to which it is associated is a Lie Ehresmann connection. Further it is shown that the Lie Ehresmann connection on adjoint bundle induced by a universal G-connection is universal over the family of adjoint bundles associated to G-bundles.

## Key words

Adjoint bundle Lie algebra bundle Lie connection Lie Ehresmann connection parallel transport universal connection

## Notes

### Acknowledgement

The second author is thankful to the SERB/DST, New Delhi, India for the financial assistance SR/S4/MS:856/13. We thank R. Rangarajan for his support.

## References

1. 1.
I. Biswas, J. Hurtubise, and J. Stasheff, A construction of a universal connection, Forum Math., 24 (2012), 365–378.
2. 2.
A. Douady and M. Lazard, Espaces fibres en algebres de Lie et en groupes, Invent. Math., 1 (1966), 133–151.
3. 3.
F. Dumitrescu, Connections and parallel transport, J. Homotopy Relat. Struct., 5(1) (2010), 171–175.
4. 4.
W. Greub, S. Halperin, and R. Vanstone, Connections, curvature and cohomology, 2, Academic press, New York (1973).
5. 5.
H. Gündoğan, Lie algebras of smooth sections, Diploma thesis, Technische Universität Darmstadt, 2007.Google Scholar
6. 6.
J. Janyška, Higher order Utiyama-like theorem, Rep. Math. Phys., 58 (2006), 93–118.
7. 7.
B. S. Kiranagi, Lie algebra bundles, Bull. Sci. Math., 2e serie, 102 (1978), 57–62.
8. 8.
B. S. Kiranagi, Lie algebra bundles and Lie rings, Proc. Nat. Acad. Sci. India, 54(A), I, (1984), 38–44.
9. 9.
K. C. H. Mackenzie, General theory of Lie groupoids and Lie algebroids, Cambridge University Press (2005).
10. 10.
M. S. Narasimhan and S. Ramanan, Existence of universal connections, Amer. J. Math., 83 (1961), 563–572.
11. 11.
M. S. Narasimhan and S. Ramanan, Existence of universal connections II, Amer. J. Math., 85 (1963), 223–231.
12. 12.
R. Schlafly, Universal connections, Invent. Math., 59 (1980), 59–65.
13. 13.
M. Spivak, A comprehensive introduction to differential geometry, 2, Third Edition, Publish or Perish INC., Houston (1999).
14. 14.
F. H. Vasilescu, Normed Lie algebras, Can. J. Math., XXIV(4) (1972), 580–591.