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

• Lie coalgebra
• Manin triple

References

• H.-D. Doebner, J.-D. Hennig, eds, Quantum groups, Proceedings of the 8th International Workshop on Mathematical Physics, Arnold Sommerfeld Institute, Claausthal, FRG, 1989, Springer-Verlag Berlin, ISBN 3-540-53503-9.
• Vyjayanthi Chari and Andrew Pressley, A Guide to Quantum Groups, (1994), Cambridge University Press, Cambridge ISBN 0-521-55884-0.
• Beisert, N., & Spill, F. (2009). The classical r-matrix of AdS/CFT and its Lie bialgebra structure. Communications in mathematical physics, 285(2), 537-565. dx.doi.org/10.1007/s00220-008-0578-2