Lie bialgebra

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.

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 cocyle 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}_XY=[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.

See also

References