Lie bialgebroid


In differential geometry, a field in mathematics, a Lie bialgebroid consists of two compatible Lie algebroids defined on dual vector bundles. Lie bialgebroids are the vector bundle version of Lie bialgebras.

Definition

Preliminary notions

A Lie algebroid consists of a bilinear skew-symmetric operation on the sections of a vector bundle ' over a smooth manifold ', together with a vector bundle morphism ' subject to the Leibniz rule
and Jacobi identity
where ' are sections of ' and ' is a smooth function on '.
The Lie bracket ' can be extended to multivector fields ' graded symmetric via the Leibniz rule
for homogeneous multivector fields '.
The Lie algebroid differential is an -linear operator ' on the '-forms ' of degree 1 subject to the Leibniz rule
for '-forms ' and '. It is uniquely characterized by the conditions
and
for functions ' on ', '-1-forms ' and ' sections of '.

The definition

A Lie bialgebroid consists of two Lie algebroids ' and ' on the dual vector bundles ' and ', subject to the compatibility
for all sections ' of '. Here ' denotes the Lie algebroid differential of ' which also operates on the multivector fields .

Symmetry of the definition

It can be shown that the definition is symmetric in ' and ', i.e. ' is a Lie bialgebroid if and only if ' is.

Examples

  1. A Lie bialgebra consists of two Lie algebras ' and ' on dual vector spaces and ' such that the Chevalley–Eilenberg differential ' is a derivation of the -bracket.
  2. A Poisson manifold ' gives naturally rise to a Lie bialgebroid on ' and '. The '-differential is and the compatibility follows then from the Jacobi identity of the Schouten bracket.

Infinitesimal version of a Poisson groupoid

It is well known that the infinitesimal version of a Lie groupoid is a Lie algebroid. Therefore, one can ask which structures need to be differentiated in order to obtain a Lie bialgebroid.

Definition of Poisson groupoid

A Poisson groupoid is a Lie groupoid ' together with a Poisson structure ' on ' such that the graph ' of the multiplication map is coisotropic. An example of a Poisson-Lie groupoid is a Poisson-Lie group. Another example is a symplectic groupoid.

Differentiation of the structure

Remember the construction of a Lie algebroid from a Lie groupoid. We take the '-tangent fibers and consider their vector bundle pulled back to the base manifold '. A section of this vector bundle can be identified with a '-invariant '-vector field on ' which form a Lie algebra with respect to the commutator bracket on '.
We thus take the Lie algebroid ' of the Poisson groupoid. It can be shown that the Poisson structure induces a fiber-linear Poisson structure on '. Analogous to the construction of the cotangent Lie algebroid of a Poisson manifold there is a Lie algebroid structure on ' induced by this Poisson structure. Analogous to the Poisson manifold case one can show that ' and form a Lie bialgebroid.

Double of a Lie bialgebroid and superlanguage of Lie bialgebroids

For Lie bialgebras ' there is the notion of Manin triples, i.e. ' can be endowed with the structure of a Lie algebra such that and ' are subalgebras and ' contains the representation of ' on ', vice versa. The sum structure is just

Courant algebroids

It turns out that the naive generalization to Lie algebroids does not give a Lie algebroid any more. Instead one has to modify either the Jacobi identity or violate the skew-symmetry and is thus lead to Courant algebroids.

Superlanguage

The appropriate superlanguage of a Lie algebroid ' is ', the supermanifold whose space of functions are the '-forms. On this space the Lie algebroid can be encoded via its Lie algebroid differential, which is just an odd vector field.
As a first guess the super-realization of a Lie bialgebroid ' should be '. But unfortunately ' is not a differential, basically because ' is not a Lie algebroid. Instead using the larger N-graded manifold ' to which we can lift ' and ' as odd Hamiltonian vector fields, then their sum squares to ' iff ' is a Lie bialgebroid.