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

Example

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

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

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

$⟨(X_{-},X_{+}),Y⟩:=K(X_{+}-X_{-},Y)$

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

Relation to Poisson–Lie groups

The Lie algebra $\mathfrak{𝔤}$ of a Poisson–Lie group G has a natural structure of Lie bialgebra. In brief the Lie group structure gives the Lie bracket on $\mathfrak{𝔤}$ as usual, and the linearisation of the Poisson structure on G gives the Lie bracket on ${\mathfrak{𝔤}}^{\mathfrak{*}}$ (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, $\xi \in {\mathfrak{𝔤}}^{*}$. 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.
• Beisert, N.; Spill, F. (2009). "The classical r-matrix of AdS/CFT and its Lie bialgebra structure". Communications in Mathematical Physics 285 (2): 537–565. doi:10.1007/s00220-008-0578-2. Bibcode: 2009CMaPh.285..537B.