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 {\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 g ∗ {\displaystyle {\mathfrak {g}}^{*}} which is compatible. More precisely the Lie algebra structure on g {\displaystyle {\mathfrak {g}}} is given by a Lie bracket [ , ] : g ⊗ g → g {\displaystyle [\ ,\ ]:{\mathfrak {g}}\otimes {\mathfrak {g}}\to {\mathfrak {g}}} and the Lie algebra structure on g ∗ {\displaystyle {\mathfrak {g}}^{*}} is given by a Lie bracket δ ∗ : g ∗ ⊗ g ∗ → g ∗ {\displaystyle \delta ^{*}:{\mathfrak {g}}^{*}\otimes {\mathfrak {g}}^{*}\to {\mathfrak {g}}^{*}} . Then the map dual to δ ∗ {\displaystyle \delta ^{*}} is called the cocommutator,
δ : g → g ⊗ g {\displaystyle \delta :{\mathfrak {g}}\to {\mathfrak {g}}\otimes {\mathfrak {g}}}
and the compatibility condition is the following cocycle relation:
δ ( [ X , Y ] ) = ( ad X ⊗ 1 + 1 ⊗ ad X ) δ ( Y ) − ( ad Y ⊗ 1 + 1 ⊗ ad Y ) δ ( X ) {\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 ad X Y = [ X , Y ] {\displaystyle \operatorname {ad} _{X}Y=[X,Y]} is the adjoint. Note that this definition is symmetric and g ∗ {\displaystyle {\mathfrak {g}}^{*}} is also a Lie bialgebra, the dual Lie bialgebra.
Example Let g {\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 t ⊂ g {\displaystyle {\mathfrak {t}}\subset {\mathfrak {g}}} and a choice of positive roots. Let b ± ⊂ g {\displaystyle {\mathfrak {b}}_{\pm }\subset {\mathfrak {g}}} be the corresponding opposite Borel subalgebras, so that t = b − ∩ b + {\displaystyle {\mathfrak {t}}={\mathfrak {b}}_{-}\cap {\mathfrak {b}}_{+}} and there is a natural projection π : b ± → t {\displaystyle \pi :{\mathfrak {b}}_{\pm }\to {\mathfrak {t}}} . Then define a Lie algebra
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