In mathematics, a Lie algebroid is a vector bundle A → M {\displaystyle A\rightarrow M} together with a Lie bracket on its space of sections Γ ( A ) {\displaystyle \Gamma (A)} and a vector bundle morphism ρ : A → T M {\displaystyle \rho :A\rightarrow TM} , satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra. Lie algebroids play a similar same role in the theory of Lie groupoids that Lie algebras play in the theory of Lie groups: reducing global problems to infinitesimal ones. Indeed, any Lie groupoid gives rise to a Lie algebroid, which is the vertical bundle of the source map restricted at the units. However, unlike Lie algebras, not every Lie algebroid arises from a Lie groupoid. Lie algebroids were introduced in 1967 by Jean Pradines.
Definition and basic concepts A Lie algebroid is a triple ( A , [ ⋅ , ⋅ ] , ρ ) {\displaystyle (A,[\cdot ,\cdot ],\rho )} consisting of
a vector bundle A {\displaystyle A} over a manifold M {\displaystyle M}
a Lie bracket [ ⋅ , ⋅ ] {\displaystyle [\cdot ,\cdot ]} on its space of sections Γ ( A ) {\displaystyle \Gamma (A)}
a morphism of vector bundles ρ : A → T M {\displaystyle \rho :A\rightarrow TM} , called the anchor, where T M {\displaystyle TM} is the tangent bundle of M {\displaystyle M}
such that the anchor and the bracket satisfy the following Leibniz rule:
[ X , f Y ] = ρ ( X ) f ⋅ Y + f [ X , Y ] {\displaystyle [X,fY]=\rho (X)f\cdot Y+f[X,Y]}
where X , Y ∈ Γ ( A ) , f ∈ C ∞ ( M ) {\displaystyle X,Y\in \Gamma (A),f\in C^{\infty }(M)} . Here ρ ( X ) f {\displaystyle \rho (X)f} is the image of f {\displaystyle f} via the derivation ρ ( X ) {\displaystyle \rho (X)} , i.e. the Lie derivative of f {\displaystyle f} along the vector field ρ ( X ) {\displaystyle \rho (X)} . The notation ρ ( X ) f ⋅ Y {\displaystyle \rho (X)f\cdot Y} denotes the (point-wise) product between the function ρ ( X ) f {\displaystyle \rho (X)f} and the vector field Y {\displaystyle Y} . One often writes A → M {\displaystyle A\to M} when the bracket and the anchor are clear from the context; some authors denote Lie algebroids by A ⇒ M {\displaystyle A\Rightarrow M} , suggesting a "limit" of a Lie groupoids when the arrows denoting source and target become "infinitesimally close".
First properties It follows from the definition that
for every x ∈ M {\displaystyle x\in M} , the kernel g x ( A ) = ker ( ρ x ) {\displaystyle {\mathfrak {g}}_{x}(A)=\ker(\rho _{x})} is a Lie algebra, called the isotropy Lie algebra at x {\displaystyle x}
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