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 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

g ′ := { ( X − , X + ) ∈ b − × b + | π ( X − ) + π ( X + ) = 0 } {\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 b − × b + {\displaystyle {\mathfrak {b}}_{-}\times {\mathfrak {b}}_{+}}, and has the same dimension as g {\displaystyle {\mathfrak {g}}}. Now identify g ′ {\displaystyle {\mathfrak {g'}}} with dual of g {\displaystyle {\mathfrak {g}}} via the pairing

⟨ ( X − , X + ) , Y ⟩ := K ( X + − X − , Y ) {\displaystyle \langle (X_{-},X_{+}),Y\rangle:=K(X_{+}-X_{-},Y)}

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

Relation to Poisson–Lie groups

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

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

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

η R : G → g ⊗ g {\displaystyle \eta ^{R}:G\to {\mathfrak {g}}\otimes {\mathfrak {g}}}

The cocommutator is then the tangent map:

δ = T e η R {\displaystyle \delta =T_{e}\eta ^{R}\,}

so that

[ ξ 1 , ξ 2 ] = δ ∗ ( ξ 1 ⊗ ξ 2 ) {\displaystyle [\xi _{1},\xi _{2}]=\delta ^{*}(\xi _{1}\otimes \xi _{2})}

is the dual of the cocommutator.

Lie bialgebra

Definition

Example

Relation to Poisson–Lie groups

See also