Algebraic expression

In mathematics, a Lie bialgebra is the Lie-theoretic case of a bialgebra: it's 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 $\operatorname {ad} _{X}Y=[X,Y]$ 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 $\xi =(df)_{e}$ 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 $\eta$ be the Poisson bivector on the manifold, define $\eta ^{R}$ 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
• 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, 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.