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 and the compatibility condition is the following cocycle relation:

is also a Lie bialgebra, the dual Lie bialgebra.

Example

be any semisimple Lie algebra. Then define a Lie algebra

via the pairing

is the Killing form, and is the "standard" example: it underlies the Drinfeld-Jimbo quantum group is solvable is semisimple.

Relation to Poisson–Lie groups

of a Poisson–Lie group G has a natural structure of Lie bialgebra. In brief the Lie group as usual, and the linearisation of the Poisson structure on G (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. The Poisson structure, as

is the Poisson bracket be the Poisson bivector 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.

See also

• Lie coalgebra

• Manin triple

References

• H.-D. Doebner, J.-D. Hennig, eds, Quantum groups, Proceedings of the 8th International Workshop on Mathematical Physics, Arnold Sommerfeld Institute, Claausthal, FRG, 1989, Springer-Verlag Berlin, .

• Vyjayanthi Chari and Andrew Pressley, A Guide to Quantum Groups, (1994), Cambridge University Press, Cambridge .