|
|
|
|
|
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 ![$ [~,~]$](http://images.planetmath.org:8080/cache/objects/10816/l2h/img6.png "$ [~,~]$") on the space of sections  , and a bundle map
 , usually called the anchor. Furthermore, there is an induced map
 , which is required to be a map of Lie algebras, such that given sections
 and a differentiable function  , the following Leibniz rule is satisfied :
![$\displaystyle [ \alpha , f \beta] = f [\alpha , \beta] + (\Upsilon (\alpha )) \beta~.$](http://images.planetmath.org:8080/cache/objects/10816/l2h/img12.png "$\displaystyle [ \alpha , f \beta] = f [\alpha , \beta] + (\Upsilon (\alpha )) \beta~.$") |
(0.1) |
Example. A typical example of a Lie algebroid is obtained when  is a Poisson manifold and  , that is  is the cotangent bundle of  .
Now suppose we have a Lie groupoid
 :
![$\displaystyle r,s~:~ \xymatrix{ \mathsf{G} \ar@<1ex>[r]^r \ar[r]_s & \mathsf{G}^{(0)}}=M~.$](http://images.planetmath.org:8080/cache/objects/10816/l2h/img18.png "$\displaystyle r,s~:~ \xymatrix{ \mathsf{G} \ar@<1ex>[r]^r \ar[r]_s & \mathsf{G}^{(0)}}=M~.$") |
(0.2) |
There is an associated Lie algebroid
 , which in the guise of a vector bundle, it is the restriction to  of the bundle of tangent vectors along the fibers of  (ie. the  -vertical vector fields). Also, the space of sections
 can be identified with the space of  -vertical, right-invariant vector fields
 which can be seen to be closed under ![$ [~,~]$](http://images.planetmath.org:8080/cache/objects/10816/l2h/img26.png "$ [~,~]$") , 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
 .
Remarks 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).
|
"Lie algebroids" is owned by bci1.
|
|
(view preamble | get metadata)
See Also: Hamiltonian algebroids, tangent bundle, Hamiltonian operator, algebroid structures and extended symmetries, Lie superalgebra
| Other names: |
extended quantum symmetry structures, generalized double algebras |
| Also defines: |
Lie algebroid, anchor, bundle map |
| Keywords: |
double algebras, extensions of Lie algebras, Lie groupoids, 'Weinstein' groupoids, extended quantum symmetry structures |
|
|
Cross-references: quantum gravity theories, developments, Hamiltonian algebroids, relation, subset, Lie groups, operation, induces, closed under, fibers, tangent vectors, restriction, Lie groupoid, cotangent bundle, Leibniz rule, differentiable function, map, induced, sections, Lie bracket, vector bundle, vector fields, manifold, geometry, infinitesimal, site, groupoid, curves, equivalence class, point, tangent space, tangent bundle, symmetries, represent, quantum systems, Lie algebras
There are 4 references to this entry.
This is version 18 of Lie algebroids, born on 2008-07-18, modified 2008-08-24.
Object id is 10816, canonical name is LieAlgebroids.
Accessed 628 times total.
Classification:
| AMS MSC: | 81R05 (Quantum theory :: Groups and algebras in quantum theory :: Finite-dimensional groups and algebras motivated by physics and their representations) | | | 81R50 (Quantum theory :: Groups and algebras in quantum theory :: Quantum groups and related algebraic methods) | | | 81R10 (Quantum theory :: Groups and algebras in quantum theory :: Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $W$-algebras and other current alg) | | | 81R15 (Quantum theory :: Groups and algebras in quantum theory :: Operator algebra methods) | | | 81P05 (Quantum theory :: Axiomatics, foundations, philosophy :: General and philosophical) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
