In mathematics, a Lie bialgebra is the Lie-theoretic case of a bialgebra: it's a set with a Lie algebra and a Lie coalgebra structure which are compatible.
It is a bialgebra where the comultiplication 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.
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 ∗ which is compatible. More precisely the Lie algebra structure on g is given by a Lie bracket [ , ] : g ⊗ g → g and the Lie algebra structure on g ∗ is given by a Lie bracket δ ∗ : g ∗ ⊗ g ∗ → g ∗ . Then the map dual to δ ∗ is called the cocommutator, δ : g → g ⊗ g and the compatibility condition is the following cocyle relation:
δ ( [ X , Y ] ) = ( ad X ⊗ 1 + 1 ⊗ ad X ) δ ( Y ) − ( ad Y ⊗ 1 + 1 ⊗ ad Y ) δ ( X ) where ad X Y = [ X , Y ] is the adjoint. Note that this definition is symmetric and g ∗ is also a Lie bialgebra, the dual Lie bialgebra.
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 ⊂ g and a choice of positive roots. Let b ± ⊂ g be the corresponding opposite Borel subalgebras, so that t = b − ∩ b + and there is a natural projection π : b ± → t . Then define a Lie algebra
g ′ := { ( X − , X + ) ∈ b − × b + | π ( X − ) + π ( X + ) = 0 } which is a subalgebra of the product b − × b + , and has the same dimension as g . Now identify g ′ with dual of g via the pairing
⟨ ( X − , X + ) , Y ⟩ := K ( X + − X − , Y ) where Y ∈ 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 ′ is solvable, whereas g is semisimple.
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 ∗ (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 ∈ C ∞ ( G ) being two smooth functions on the group manifold. Let ξ = ( d f ) e be the differential at the identity element. Clearly, ξ ∈ g ∗ . The Poisson structure on the group then induces a bracket on g ∗ , as
[ ξ 1 , ξ 2 ] = ( d { f 1 , f 2 } ) e where { , } is the Poisson bracket. Given η be the Poisson bivector on the manifold, define η R to be the right-translate of the bivector to the identity element in G. Then one has that
η R : G → g ⊗ g The cocommutator is then the tangent map:
δ = T e η R so that
[ ξ 1 , ξ 2 ] = δ ∗ ( ξ 1 ⊗ ξ 2 ) is the dual of the cocommutator.
