Lie Algebroid - Lie Algebroid Associated To A Lie Groupoid

Lie Algebroid Associated To A Lie Groupoid

To describe the construction let us fix some notation. G is the space of morphisms of the Lie groupoid, M the space of objects, the units and the target map.

the t-fiber tangent space. The Lie algebroid is now the vector bundle . 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 . 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 . The target map is and the units . The t-fibers are and therefore . So the Lie algebroid is the vector bundle . The extension of sections X into A to left-invariant vector fields on G is simply and the extension of a smooth function f from M to a left-invariant function on G is . 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, where is the inverse map.