Remember that a Lie algebroid is defined as a skew-symmetric operation on the sections Γ of a vector bundle A→M over a smooth manifoldM together with a vector bundle morphismρ: A→TM subject to the Leibniz rule and Jacobi identity where Φ, ψk are sections of A and f is a smooth function on M. The Lie bracketA can be extended to multivector fields Γ graded symmetric via the Leibniz rule for homogeneous multivector fields Φ, Ψ, Χ. The Lie algebroid differential is an R-linear operator dA on the A-forms ΩA = Γ of degree 1 subject to the Leibniz-rule for A-forms α and β. It is uniquely characterized by the conditions and for functions f on M, A-1-forms α∈Γ and Φ, ψ sections of A.
The definition
A Lie bialgebroid are two Lie algebroids and on dual vector bundlesA→M and A*→M subject to the compatibility for all sections Φ, ψ of A. Here d* denotes the Lie algebroid differential of A* which also operates on the multivector fields Γ.
Symmetry of the definition
It can be shown that the definition is symmetric in A and A*, i.e. is a Lie bialgebroid iff is.
Examples
1. A Lie bialgebra are two Lie algebras and on dual vector spacesg and g* such that the Chevalley–Eilenberg differential δ* is a derivation of the g-bracket. 2. A Poisson manifold gives naturally rise to a Lie bialgebroid on TM and T*M with the Lie bracket induced by the Poisson structure. The T*M-differential is d*= 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 π on G such that the multiplication graph m ⊂ G×G× 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 t-tangent fibers and consider their vector bundle pulled back to the base manifoldM. A section of this vector bundle can be identified with a G-invariant t-vector field on G which form a Lie algebrawith respect to the commutator bracket on TG. We thus take the Lie algebroid A→M of the Poisson groupoid. It can be shown that the Poisson structure induces a fiber-linear Poisson structure on A. Analogous to the construction of the cotangent Lie algebroid of a Poisson manifold there is a Lie algebroid structure on A* induced by this Poisson structure. Analogous to the Poisson manifold case one can show that A and A* form a Lie bialgebroid.
Double of a Lie bialgebroid and superlanguage of Lie bialgebroids
For Lie bialgebroids there is the notion of Manin triples, i.e. c=g+g* can be endowed with the structure of a Lie algebra such that g and g* are subalgebras and c contains the representation of g on g*, 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 A is ΠA, the supermanifold whose space of functions are the A-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 ΠA+ΠA*. But unfortunately dA +d*|ΠA+ΠA* is not a differential, basically because A+A* is not a Lie algebroid. Instead using the larger N-graded manifoldT*A = T*A* to which we can lift dA and d* as odd Hamiltonian vector fields, then their sum squares to 0 iff is a Lie bialgebroid.
