Lie bialgebra

In mathematics, a Lie bialgebra is the Lie-theoretic case of a bialgebra: it is a set with a Lie algebra and a Lie coalgebra structure which are compatible.
It is a bialgebra where the multiplication is skew-symmetric and satisfies a dual Jacobi identity, so that the dual vector space is a Lie algebra, whereas the comultiplication is a 1-cocycle, so that the multiplication and comultiplication are compatible. The cocycle condition implies that, in practice, one studies only classes of bialgebras that are cohomologous to a Lie bialgebra on a coboundary.
They are also called Poisson-Hopf algebras, and are the Lie algebra of a Poisson–Lie group.
Lie bialgebras occur naturally in the study of the Yang–Baxter equations.
Definition.
A vector space formula_1 is a Lie bialgebra if it is a Lie algebra,
and there is the structure of Lie algebra also on the dual vector space formula_2 which is compatible.
More precisely the Lie algebra structure on formula_1 is given 
by a Lie bracket formula_4 
and the Lie algebra structure on formula_2 is given by a Lie
bracket formula_6.
Then the map dual to formula_7 is called the cocommutator, 
formula_8
and the compatibility condition is the following cocycle relation:
where formula_10 is the adjoint.
Note that this definition is symmetric and formula_2 is also a Lie bialgebra, the dual Lie bialgebra.
Example.
Let formula_1 be any semisimple Lie algebra. 
To specify a Lie bialgebra structure we thus need to specify a compatible Lie algebra structure on the dual vector space. 
Choose a Cartan subalgebra formula_13 and a choice of positive roots. 
Let formula_14 be the corresponding opposite Borel subalgebras, so that formula_15 and there is a natural projection formula_16.
Then define a Lie algebra 
which is a subalgebra of the product formula_18, and has the same dimension as formula_1.
Now identify formula_20 with dual of formula_1 via the pairing
where formula_23 and formula_24 is the Killing form.
This defines a Lie bialgebra structure on formula_1, and is the "standard" example: it underlies the Drinfeld-Jimbo quantum group.
Note that formula_20 is solvable, whereas formula_1 is semisimple.
Relation to Poisson-Lie groups.
The Lie algebra formula_1 of a Poisson-Lie group "G" has a natural structure of Lie bialgebra.
In brief the Lie group structure gives the Lie bracket on formula_1 as usual, and the linearisation of the Poisson structure on "G" 
gives the Lie bracket on 
formula_30 (recalling that a linear Poisson structure on a vector space is the same thing as a Lie bracket on the dual vector space). 
In more detail, let "G" be a Poisson-Lie group, with formula_31 being two smooth functions on the group manifold. Let formula_32 be the differential at the identity element. Clearly, formula_33. The Poisson structure on the group then induces a bracket on formula_2, as
where formula_36 is the Poisson bracket. Given formula_37 be the Poisson bivector on the manifold, define formula_38 to be the right-translate of the bivector to the identity element in "G". Then one has that
The cocommutator is then the tangent map:
so that
is the dual of the cocommutator.