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

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

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

Example

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

${\displaystyle {\mathfrak {g')):=\{(X_{-},X_{+})\in {\mathfrak {b))_{-}\times {\mathfrak {b))_{+}\ {\bigl \vert }\ \pi (X_{-})+\pi (X_{+})=0\))$

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

${\displaystyle \langle (X_{-},X_{+}),Y\rangle :=K(X_{+}-X_{-},Y)}$

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

Relation to Poisson–Lie groups

The Lie algebra ${\displaystyle {\mathfrak {g))}$ of a Poisson–Lie group G has a natural structure of Lie bialgebra. In brief the Lie group structure gives the Lie bracket on ${\displaystyle {\mathfrak {g))}$ as usual, and the linearisation of the Poisson structure on G gives the Lie bracket on ${\displaystyle {\mathfrak {g^{*))))$ (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 ${\displaystyle f_{1},f_{2}\in C^{\infty }(G)}$ being two smooth functions on the group manifold. Let ${\displaystyle \xi =(df)_{e))$ be the differential at the identity element. Clearly, ${\displaystyle \xi \in {\mathfrak {g))^{*))$. The Poisson structure on the group then induces a bracket on ${\displaystyle {\mathfrak {g))^{*))$, as

${\displaystyle [\xi _{1},\xi _{2}]=(d\{f_{1},f_{2}\})_{e}\,}$

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

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

The cocommutator is then the tangent map:

${\displaystyle \delta =T_{e}\eta ^{R}\,}$

so that

${\displaystyle [\xi _{1},\xi _{2}]=\delta ^{*}(\xi _{1}\otimes \xi _{2})}$

is the dual of the cocommutator.