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 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 group G 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 (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