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.
Lie bialgebras occur naturally in the study of the Yang-Baxter equations.
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 cocyle relation:
where is the adjoint. Note that this definition is symmetric and is also a Lie bialgebra, the dual Lie bialgebra.
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 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
The cocommutator is then the tangent map:
is the dual of the cocommutator.
- 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, ISBN 3-540-53503-9.
- Vyjayanthi Chari and Andrew Pressley, A Guide to Quantum Groups, (1994), Cambridge University Press, Cambridge ISBN 0-521-55884-0.