Lie bialgebra

In mathematics, a Lie bialgebra is the Lie-theoretic case of a bialgebra: its 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

More precisely, comultiplication on the algebra, $\delta:\mathfrak{g} \to \mathfrak{g} \otimes \mathfrak{g}$, is called the cocommutator, and must satisfy two properties. The dual

$\delta^*:\mathfrak{g}^* \otimes \mathfrak{g}^* \to \mathfrak{g}^*$

must be a Lie bracket on $\mathfrak{g}^*$, and it must be a cocycle:

$\delta([X,Y]) = \left( \operatorname{ad}_X \otimes 1 + 1 \otimes \operatorname{ad}_X \right) \delta(Y) - \left( \operatorname{ad}_Y \otimes 1 + 1 \otimes \operatorname{ad}_Y \right) \delta(X)$

where $\operatorname{ad}_XY=[X,Y]$ is the adjoint.

Relation to Poisson-Lie groups

Let G be a Poisson-Lie group, with $f_1,f_2 \in C^\infty(G)$ being two smooth functions on the group manifold. Let ξ = (df)_e be the differential at the identity element. Clearly, $\xi \in \mathfrak{g}^*$. The Poisson structure on the group then induces a bracket on $\mathfrak{g}^*$, as

$[\xi_1,\xi_2]=(d\{f_1,f_2\})_e\,$

where {,} is the Poisson bracket. Given η be the Poisson bivector on the manifold, define η^R to be the right-translate of the bivector to the identity element in G. Then one has that

$\eta^R:G\to \mathfrak{g} \otimes \mathfrak{g}$

The cocommutator is then the tangent map:

$\delta = T_e \eta^R\,$

so that

$[\xi_1,\xi_2]= \delta^*(\xi_1 \otimes \xi_2)$

is the dual of the cocommutator.

See also

• Lie coalgebra

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, 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.