Different characterizations of curvature in the context of Lie algebroids
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We consider a vector bundle map $F\colon E_{1}\raw E_{2}$ between Lie algebroids $E_{1}$ and $E_{2}$ over arbitrary bases $M_{1}$ and $M_{2}$. We associate to it different notions of curvature which we call A-curvature, Q-curvature, P-curvature, and S-curvature using the different characterizations of Lie algebroid structure, namely Lie algebroid, Q-manifold, Poisson and Schouten structures. We will see that these curvatures generalize the ordinary notion of curvature defined for a vector bundle, and we will prove that these curvatures are equivalent, in the sense that $F$ is a morphism of Lie algebroids if and only if one (and hence all) of these curvatures is null. In particular we get as a corollary that $F$ is a morphism of Lie algebroids if and only if the corresponding map is a morphism of Poisson manifolds (resp. Schouten supermanifolds).
Abstract
We consider a vector bundle map $F\colon E_{1}\raw E_{2}$ between Lie algebroids $E_{1}$ and $E_{2}$ over arbitrary bases $M_{1}$ and $M_{2}$. We associate to it different notions of curvature which we call A-curvature, Q-curvature, P-curvature, and S-curvature using the different characterizations of Lie algebroid structure, namely Lie algebroid, Q-manifold, Poisson and Schouten structures. We will see that these curvatures generalize the ordinary notion of curvature defined for a vector bundle, and we will prove that these curvatures are equivalent, in the sense that $F$ is a morphism of Lie algebroids if and only if one (and hence all) of these curvatures is null. In particular we get as a corollary that $F$ is a morphism of Lie algebroids if and only if the corresponding map is a morphism of Poisson manifolds (resp. Schouten supermanifolds).
참고문헌 (10)
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