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 $\mathfrak$ is a Lie bialgebra if it is a Lie algebra,
and there is the structure of Lie algebra also on the dual vector space $\mathfrak^*$ which is compatible.
More precisely the Lie algebra structure on $\mathfrak$ is given
by a Lie bracket ,\ \mathfrak \otimes \mathfrak \to \mathfrak
and the Lie algebra structure on $\mathfrak^*$ is given by a Lie
bracket $\delta^*:\mathfrak^* \otimes \mathfrak^* \to \mathfrak^*$.
Then the map dual to $\delta^*$ is called the cocommutator,
$\delta:\mathfrak \to \mathfrak \otimes \mathfrak$
and the compatibility condition is the following cocycle relation:

:\delta( ,Y = \left(\operatorname_X \otimes 1 + 1 \otimes \operatorname_X\right) \delta(Y) - \left(\operatorname_Y \otimes 1 + 1 \otimes \operatorname_Y\right) \delta(X)

where \operatorname_XY= ,Y/math> is the adjoint.
Note that this definition is symmetric and $\mathfrak^*$ is also a Lie bialgebra, the dual Lie bialgebra.

Example

Let $\mathfrak$ 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 $\mathfrak\subset \mathfrak$ and a choice of positive roots.
Let $\mathfrak_\pm\subset \mathfrak$ be the corresponding opposite Borel subalgebras, so that $\mathfrak = \mathfrak_-\cap\mathfrak_+$ and there is a natural projection $\pi:\mathfrak_\pm \to \mathfrak$.
Then define a Lie algebra
:$\mathfrak := \$
which is a subalgebra of the product $\mathfrak_- \times \mathfrak_+$, and has the same dimension as $\mathfrak$.
Now identify $\mathfrak$ with dual of $\mathfrak$ via the pairing
:$\langle (X_-,X_+), Y \rangle := K(X_+ - X_-, Y)$
where $Y\in \mathfrak$ and $K$ is the Killing form.
This defines a Lie bialgebra structure on $\mathfrak$, and is the "standard" example: it underlies the Drinfeld-Jimbo quantum group.
Note that $\mathfrak$ is solvable, whereas $\mathfrak$ is semisimple.

Relation to Poisson–Lie groups

The Lie algebra $\mathfrak$ of a Poisson–Lie group ''G'' has a natural structure of Lie bialgebra.
In brief the Lie group structure gives the Lie bracket on $\mathfrak$ as usual, and the linearisation of the Poisson structure on ''G''
gives the Lie bracket on
$\mathfrak$ (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, $\xi \in \mathfrak^*$. The Poisson structure on the group then induces a bracket on $\mathfrak^*$, as

: xi_1,\xi_2= (d\)_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

:$\eta^R:G\to \mathfrak \otimes \mathfrak$

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, .
* Vyjayanthi Chari and Andrew Pressley, ''A Guide to Quantum Groups'', (1994), Cambridge University Press, Cambridge .
* {{cite journal , last1 = Beisert , first1 = N. , last2 = Spill , first2 = F. , year = 2009 , title = The classical r-matrix of AdS/CFT and its Lie bialgebra structure , journal = Communications in Mathematical Physics , volume = 285 , issue = 2, pages = 537–565 , doi = 10.1007/s00220-008-0578-2 , arxiv = 0708.1762 , bibcode = 2009CMaPh.285..537B , s2cid = 8946457

Lie algebras
Coalgebras
Symplectic geometry