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.
is a Lie bialgebra if it is a Lie algebra, and there is the structure of Lie algebra also on the dual vector space
More precisely the Lie algebra structure on
and the Lie algebra structure on
is called the cocommutator,
and the compatibility condition is the following cocycle relation: where
Note that this definition is symmetric and
is also a Lie bialgebra, the dual Lie bialgebra.
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.
be the corresponding opposite Borel subalgebras, so that
Then define a Lie algebra which is a subalgebra of the product
is the Killing form.
This defines a Lie bialgebra structure on
, and is the "standard" example: it underlies the Drinfeld-Jimbo quantum group.
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).
being two smooth functions on the group manifold.
be the differential at the identity element.
The Poisson structure on the group then induces a bracket on
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.