Lie algebroid

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 space 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:

${\displaystyle [X,fY]=\rho (X)f\cdot Y+f[X,Y]}$

where ${\displaystyle X,Y\in \Gamma (E),f\in C^{\infty }(M)}$ and ${\displaystyle \rho (X)f}$ is the derivative of ${\displaystyle f}$ along the vector field ${\displaystyle \rho (X)}$. It follows that

${\displaystyle \rho ([X,Y])=[\rho (X),\rho (Y)]}$

for all ${\displaystyle X,Y\in \Gamma (E)}$.

• Every Lie algebra is a Lie algebroid over the one point manifold.
• The tangent bundle ${\displaystyle TM}$ of a manifold ${\displaystyle M}$ is a Lie algebroid for the Lie bracket of vector fields and the identity of ${\displaystyle TM}$ as an anchor.
• Every integrable subbundle of the tangent bundle — that is, one whose sections are closed under the Lie bracket — also defines a Lie algebroid.
• Every bundle of Lie algebras over a smooth manifold defines a Lie algebroid where the Lie bracket is defined pointwise and the anchor map is equal to zero.
• To every Lie groupoid is associated a Lie algebroid, generalizing how a Lie algebra is associated to a Lie group (see also below). For example, the Lie algebroid ${\displaystyle TM}$ comes from the pair groupoid whose objects are ${\displaystyle M}$, with one isomorphism between each pair of objects. Unfortunately, going back from a Lie algebroid to a Lie groupoid is not always possible,[1] but every Lie algebroid gives a stacky Lie groupoid.[2][3]
• Given the action of a Lie algebra g on a manifold M, the set of g -invariant vector fields on M is a Lie algebroid over the space of orbits of the action.
• The Atiyah algebroid of a principal G-bundle P over a manifold M is a Lie algebroid with short exact sequence:

  ${\displaystyle 0\to P\times _{G}{\mathfrak {g}}\to TP/G{\xrightarrow {\rho }}TM\to 0.}$

The space of sections of the Atiyah algebroid is the Lie algebra of G-invariant vector fields on P.

• A Poisson Lie algebroid is associated to a Poisson manifold by taking E to be the cotangent bundle. The anchor map is given by the Poisson bivector. This can be seen in a Lie bialgebroid.

To describe the construction let us fix some notation. G is the space of morphisms of the Lie groupoid, M the space of objects, ${\displaystyle e:M\to G}$ the units and ${\displaystyle t:G\to M}$ the target map.

${\displaystyle T^{t}G=\bigcup _{p\in M}T(t^{-1}(p))\subset TG}$ the t-fiber tangent space. The Lie algebroid is now the vector bundle ${\displaystyle A:=e^{*}T^{t}G}$. This inherits a bracket from G, because we can identify the M-sections into A with left-invariant vector fields on G. The anchor map then is obtained as the derivation of the source map ${\displaystyle Ts:e^{*}T^{t}G\rightarrow TM}$. Further these sections act on the smooth functions of M by identifying these with left-invariant functions on G.

As a more explicit example consider the Lie algebroid associated to the pair groupoid ${\displaystyle G:=M\times M}$. The target map is ${\displaystyle t:G\to M:(p,q)\mapsto p}$ and the units ${\displaystyle e:M\to G:p\mapsto (p,p)}$. The t-fibers are ${\displaystyle p\times M}$ and therefore ${\displaystyle T^{t}G=\bigcup _{p\in M}p\times TM\subset TM\times TM}$. So the Lie algebroid is the vector bundle ${\displaystyle A:=e^{*}T^{t}G=\bigcup _{p\in M}T_{p}M=TM}$. The extension of sections X into A to left-invariant vector fields on G is simply ${\displaystyle {\tilde {X}}(p,q)=0\oplus X(q)}$ and the extension of a smooth function f from M to a left-invariant function on G is ${\displaystyle {\tilde {f}}(p,q)=f(q)}$. Therefore, the bracket on A is just the Lie bracket of tangent vector fields and the anchor map is just the identity.

Of course you could do an analog construction with the source map and right-invariant vector fields/ functions. However you get an isomorphic Lie algebroid, with the explicit isomorphism ${\displaystyle i_{*}}$, where ${\displaystyle i:G\to G}$ is the inverse map.

• R-algebroid