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 (thus suggesting a groupoid
approach), as well as serving as a site on which to study infinitesimal geometry (see, for example, ref. [1]). The formal definition of a Lie algebroid is presented next.
Definition 0.1
Let

be a manifold and let

denote the set of
vector fields
on

. Then, a
Lie 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 following
Leibniz rule is satisfied :
![$\displaystyle [ \alpha , f \beta] = f [\alpha , \beta] + (\Upsilon (\alpha )) \beta~.$](img12.png) |
(0.1) |
Example 0.1
A typical example of a Lie algebroid is obtained when

is a Poisson
manifold and

, that is

is the cotangent bundle of

.
Now suppose we have a Lie groupoid
:
![$\displaystyle r,s~:~ \xymatrix{ \mathsf{G} \ar@<1ex>[r]^r \ar[r]_s & \mathsf{G}^{(0)}}=M~.$](img18.png) |
(0.2) |
There is an associated Lie algebroid
, which in the
guise of a vector bundle, it is the restriction to
of the
bundle of tangent vectors along the fibers of
(ie. the
-vertical vector fields). Also, the space of sections
can be identified with the space of
-vertical,
right-invariant vector fields
which
can be seen to be closed under
, and the latter induces a
bracket operation
on
thus turning
into a Lie
algebroid. Subsequently, a Lie algebroid
is integrable if
there exists a Lie groupoid
inducing
.
Remark 0.1
Unlike Lie algebras that can be integrated to corresponding
Lie groups, not all
Lie algebroids are `smoothly integrable' to Lie groupoids; the subset of Lie groupoids that have corresponding Lie algebroids are sometimes called
`Weinstein groupoids'.
Note also the relation
of the Lie algebroids to Hamiltonian algebroids, also concerning recent developments in (relativistic) quantum gravity theories.
- 1
-
K. C. H. Mackenzie: General Theory of Lie Groupoids and Lie
Algebroids, London Math. Soc. Lecture Notes Series, 213,
Cambridge University Press: Cambridge,UK (2005).
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As of this snapshot date, this entry was owned by bci1.