# 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:
    $\delta ([X,Y])=\left(\operatorname {ad} _{X}\otimes 1+1\otimes \operatorname {ad} _{X}\right)\delta (Y)-\left(\operatorname {ad} _{Y}\otimes 1+1\otimes \operatorname {ad} _{Y}\right)\delta (X)$
where $\operatorname {ad} _{X}Y=[X,Y]$ is the adjoint. 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
    ${\mathfrak {g'}}:=\{(X_{-},X_{+})\in {\mathfrak {b}}_{-}\times {\mathfrak {b}}_{+}\ {\bigl \vert }\ \pi (X_{-})+\pi (X_{+})=0\}$
which is a subalgebra of the product ${\mathfrak {b}}_{-}\times {\mathfrak {b}}_{+}$, and has the same dimension as ${\mathfrak {g}}$. Now identify ${\mathfrak {g'}}$ with dual of ${\mathfrak {g}}$ via the pairing
    $\langle (X_{-},X_{+}),Y\rangle :=K(X_{+}-X_{-},Y)$
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 ${\mathfrak {g'}}$ 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 ${\mathfrak {g^{*}}}$ (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 $f_{1},f_{2}\in C^{\infty }(G)$ 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.