Lie algebroid

In mathematics, Lie algebroids serve the same role in the theory of Lie groupoids that Lie algebras serve in the theory of Lie groups: reducing global problems to infinitesimal ones. Just as a Lie groupoid can be thought of as a "Lie group with many objects", a Lie algebroid is like a "Lie algebra with many objects".

More precisely, a Lie algebroid is a triple ${\displaystyle (E,[\cdot ,\cdot ],\rho )}$ consisting of a vector bundle ${\displaystyle E}$ over a manifold ${\displaystyle M}$, together with a Lie bracket ${\displaystyle [\cdot ,\cdot ]}$ on its module of sections ${\displaystyle \Gamma (E)}$ and a morphism of vector bundles ${\displaystyle \rho :E\rightarrow TM}$ called the anchor. Here ${\displaystyle TM}$ is the tangent bundle of ${\displaystyle M}$. The anchor and the bracket are to satisfy the Leibniz rule:

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