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 comultiplication 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 is a Lie bialgebra if it is a Lie algebra, and there is the structure of Lie algebra also on the dual vector space which is compatible. More precisely the Lie algebra structure on is given by a Lie bracket and the Lie algebra structure on is given by a Lie bracket. Then the map dual to is called the cocommutator, and the compatibility condition is the following cocycle relation: where is the adjoint. Note that this definition is symmetric and is also a Lie bialgebra, the dual Lie bialgebra.
Example
Let 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 and a choice of positive roots. Let be the corresponding opposite Borel subalgebras, so that and there is a natural projection. Then define a Lie algebra which is a subalgebra of the product , and has the same dimension as. Now identify with dual of via the pairing where and is the Killing form. This defines a Lie bialgebra structure on, and is the "standard" example: it underlies the Drinfeld-Jimbo quantum group. Note that is solvable, whereas is semisimple.
Relation to Poisson-Lie groups
The Lie algebra of a Poisson-Lie groupG has a natural structure of Lie bialgebra. In brief the Lie group structure gives the Lie bracket on as usual, and the linearisation of the Poisson structure on G gives the Lie bracket on . In more detail, let G be a Poisson-Lie group, with being two smooth functions on the group manifold. Let be the differential at the identity element. Clearly,. The Poisson structure on the group then induces a bracket on, as where is the Poisson bracket. Given be the Poisson bivector on the manifold, define 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.
