In differential geometry, a field in mathematics, a Lie bialgebroid consists of two compatible Lie algebroids defined on dual vector bundles.
A Lie algebroid consists of a bilinear skew-symmetric operation
, together with a vector bundle morphism
subject to the Leibniz rule and Jacobi identity where
can be extended to multivector fields
graded symmetric via the Leibniz rule for homogeneous multivector fields
of degree 1 subject to the Leibniz rule for
It is uniquely characterized by the conditions and for functions
on the dual vector bundles
, subject to the compatibility for all sections
denotes the Lie algebroid differential of
which also operates on the multivector fields
It can be shown that the definition is symmetric in
Therefore, one can ask which structures need to be differentiated in order to obtain a Lie bialgebroid.
Another example is a symplectic groupoid (where the Poisson structure is non-degenerate on
-tangent fibers) and consider their vector bundle pulled back to the base manifold
A section of this vector bundle can be identified with a
which form a Lie algebra with respect to the commutator bracket on
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
Analogous to the Poisson manifold case one can show that
there is the notion of Manin triples, i.e.
can be endowed with the structure of a Lie algebra such that
The sum structure is just 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.
, the supermanifold whose space of (super)functions are the
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
Instead using the larger N-graded manifold
as odd Hamiltonian vector fields, then their sum squares to