Lie bialgebra

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 $𝔤$ is a Lie bialgebra if it is a Lie algebra, and there is the structure of Lie algebra also on the dual vector space ${𝔤}^{\ast }$ which is compatible. More precisely the Lie algebra structure on $𝔤$ is given by a Lie bracket $[,]:𝔤\otimes 𝔤\to 𝔤$ and the Lie algebra structure on ${𝔤}^{\ast }$ is given by a Lie bracket ${\delta }^{\ast }:{𝔤}^{\ast }\otimes {𝔤}^{\ast }\to {𝔤}^{\ast }$. Then the map dual to ${\delta }^{\ast }$ is called the cocommutator, $\delta :𝔤\to 𝔤\otimes 𝔤$ and the compatibility condition is the following cocycle relation:

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

where ${\mathrm{ad}}_{X}Y=[X,Y]$ is the adjoint. Note that this definition is symmetric and ${𝔤}^{\ast }$ is also a Lie bialgebra, the dual Lie bialgebra.

Example

Let $𝔤$ 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 $𝔱\subset 𝔤$ and a choice of positive roots. Let ${𝔟}_{\pm }\subset 𝔤$ be the corresponding opposite Borel subalgebras, so that $𝔱={𝔟}_{-}\cap {𝔟}_{+}$ and there is a natural projection $\pi :{𝔟}_{\pm }\to 𝔱$. Then define a Lie algebra

${𝔤}^{\prime }:=\{(X_{-},X_{+})\in {𝔟}_{-}\times {𝔟}_{+}|\pi (X_{-})+\pi (X_{+})=0\}$

which is a subalgebra of the product ${𝔟}_{-}\times {𝔟}_{+}$, and has the same dimension as $𝔤$. Now identify ${𝔤}^{\prime }$ with dual of $𝔤$ via the pairing

$⟨(X_{-},X_{+}),Y⟩:=K(X_{+}-X_{-},Y)$

where $Y\in 𝔤$ and $K$ is the Killing form. This defines a Lie bialgebra structure on $𝔤$, and is the "standard" example: it underlies the Drinfeld-Jimbo quantum group. Note that ${𝔤}^{\prime }$ is solvable, whereas $𝔤$ is semisimple.

Relation to Poisson–Lie groups

The Lie algebra $𝔤$ of a Poisson–Lie group G has a natural structure of Lie bialgebra. In brief the Lie group structure gives the Lie bracket on $𝔤$ as usual, and the linearisation of the Poisson structure on G gives the Lie bracket on ${𝔤}^{\mathfrak{\ast }}$ (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 $f_1,f_2\in C^{\mathrm{\infty }}(G)$ being two smooth functions on the group manifold. Let $\xi =(df{)}_e$ be the differential at the identity element. Clearly, $\xi \in {𝔤}^{\ast }$. The Poisson structure on the group then induces a bracket on ${𝔤}^{\ast }$, 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

${\eta }^R:G\to 𝔤\otimes 𝔤$

The cocommutator is then the tangent map:

$\delta =T_e{\eta }^R$

so that

$[{\xi }_1,{\xi }_2]={\delta }^{\ast }({\xi }_1\otimes {\xi }_2)$

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, Template:Isbn.
• Vyjayanthi Chari and Andrew Pressley, A Guide to Quantum Groups, (1994), Cambridge University Press, Cambridge Template:Isbn.
• Template:Cite journal