A note on Poisson Lie algebroids. (English) Zbl 1226.53079

Popescu, Liviu

Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 10th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 6–11, 2008. Sofia: Avangard Prima (ISBN 978-954-323-531-5/pbk). 227-236 (2009).

This paper generalizes results on Poisson manifolds to Lie algebroids. Let us recall that the notion of Lie algebroid generalizes both the concept of Lie algebra and integrable distribution. A natural example of a Lie algebroid is the cotangent bundle of a Poisson manifold. In recent years, Lie algebroids have been extensively studied by many authors, for instance by Mackenzie and Vaisman. In this work, the author studies the geometrical structures on the prolongation of a Lie algebroid to its dual bundle. More precisely, the notion of horizontal lift is introduced, and some aspects of the geometry of a Lie algebroid endowed with a Poisson structure are investigated.
Reprint of J. Geom. Symmetry Phys. 12, 63–73 (2008; Zbl 1162.53326).
For the entire collection see [Zbl 1169.81004].

Reviewer: Angela Gammella-Mathieu (Metz)

MSC:

53D17	Poisson manifolds; Poisson groupoids and algebroids
17B63	Poisson algebras
53C05	Connections (general theory)

Keywords:

Poisson manifolds; Lie algebroids; Lie algebra; dual bundle

Citations:

Zbl 1162.53326