Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 005, 29 pages      arXiv:1506.03216      https://doi.org/10.3842/SIGMA.2018.005

Poisson Geometry Related to Atiyah Sequences

Kirill Mackenzie a, Anatol Odzijewicz b and Aneta Sliżewska b
a) School of Mathematics and Statistics, University of Sheffield, Sheffield, S3 7RH, UK
b) Institute of Mathematics, University in Białystok, Ciołkowskiego 1M, 15-245 Białystok, Poland

Received July 05, 2017, in final form January 06, 2018; Published online January 10, 2018

Abstract
We construct and investigate a short exact sequence of Poisson $\mathcal{VB}$-groupoids which is canonically related to the Atiyah sequence of a $G$-principal bundle $P$. Our results include a description of the structure of the symplectic leaves of the Poisson groupoid $\frac{T^*P\times T^*P}{G}\rightrightarrows \frac{T^*P}{G}$. The semidirect product case, which is important for applications in Hamiltonian mechanics, is also discussed.

Key words: Atiyah sequence; $\mathcal{VB}$-groupoid; Poisson groupoid; dualization of $\mathcal{VB}$-groupoid.

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