Lie algebroids
0.1 Topic on Lie algebroids
This is a topic entry on Lie algebroids that focuses on their quantum applications and extensions of current algebraic theories.
Lie algebroids generalize Lie algebras, and in certain quantum systems they represent extended quantum (algebroid) symmetries. One can think of a Lie algebroid as generalizing the idea of a tangent bundle where the tangent space
at a point is effectively the equivalence class
of curves meeting at that point (thussuggesting a groupoid
approach), as well as serving as a site on which to study infinitesimal
geometry (see, for example, ref. [Mackenzie2005]). The formal definition of a Lie algebroid is presented next.
Definition 0.1Let be a manifold and let denote the set of vector fields on . Then, aLie algebroid over consists of a vector bundle
,equipped with a Lie bracket on the space of sections ,and a bundle map
, usually called the anchor.Furthermore, there is an induced map ,which is required to be a map of Lie algebras, such that given sections
and a differentiable function , the followingLeibniz rule
is satisfied :
(0.1) |
Example 0.1.
A typical example of a Lie algebroid is obtained when is a Poissonmanifold and , that is is the cotangent bundle of .
Now suppose we have a Lie groupoid :
(0.2) |