Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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DIFFERENT CHARACTERIZATIONS OF CURVATURE IN THE CONTEXT OF LIE ALGEBROIDS

  • Rabah Djabri (Department of Operations Research University of Bejaia)
  • Received : 2023.06.15
  • Accepted : 2024.05.14
  • Published : 2024.09.01
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Abstract

We consider a vector bundle map F : E1 → E2 between Lie algebroids E1 and E2 over arbitrary bases M1 and M2. 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).

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References

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