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 [math]\displaystyle{ \mathfrak{g} }[/math] is a Lie bialgebra if it is a Lie algebra, and there is the structure of Lie algebra also on the dual vector space [math]\displaystyle{ \mathfrak{g}^* }[/math] which is compatible. More precisely the Lie algebra structure on [math]\displaystyle{ \mathfrak{g} }[/math] is given by a Lie bracket [math]\displaystyle{ [\ ,\ ]:\mathfrak{g} \otimes \mathfrak{g} \to \mathfrak{g} }[/math] and the Lie algebra structure on [math]\displaystyle{ \mathfrak{g}^* }[/math] is given by a Lie bracket [math]\displaystyle{ \delta^*:\mathfrak{g}^* \otimes \mathfrak{g}^* \to \mathfrak{g}^* }[/math]. Then the map dual to [math]\displaystyle{ \delta^* }[/math] is called the cocommutator, [math]\displaystyle{ \delta:\mathfrak{g} \to \mathfrak{g} \otimes \mathfrak{g} }[/math] and the compatibility condition is the following cocycle relation:
- [math]\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) }[/math]
where [math]\displaystyle{ \operatorname{ad}_XY=[X,Y] }[/math] is the adjoint. Note that this definition is symmetric and [math]\displaystyle{ \mathfrak{g}^* }[/math] is also a Lie bialgebra, the dual Lie bialgebra.
Example
Let [math]\displaystyle{ \mathfrak{g} }[/math] 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 [math]\displaystyle{ \mathfrak{t}\subset \mathfrak{g} }[/math] and a choice of positive roots. Let [math]\displaystyle{ \mathfrak{b}_\pm\subset \mathfrak{g} }[/math] be the corresponding opposite Borel subalgebras, so that [math]\displaystyle{ \mathfrak{t} = \mathfrak{b}_-\cap\mathfrak{b}_+ }[/math] and there is a natural projection [math]\displaystyle{ \pi:\mathfrak{b}_\pm \to \mathfrak{t} }[/math]. Then define a Lie algebra
- [math]\displaystyle{ \mathfrak{g'}:=\{ (X_-,X_+)\in \mathfrak{b}_-\times\mathfrak{b}_+\ \bigl\vert\ \pi(X_-)+\pi(X_+)=0\} }[/math]
which is a subalgebra of the product [math]\displaystyle{ \mathfrak{b}_-\times\mathfrak{b}_+ }[/math], and has the same dimension as [math]\displaystyle{ \mathfrak{g} }[/math]. Now identify [math]\displaystyle{ \mathfrak{g'} }[/math] with dual of [math]\displaystyle{ \mathfrak{g} }[/math] via the pairing
- [math]\displaystyle{ \langle (X_-,X_+), Y \rangle := K(X_+-X_-,Y) }[/math]
where [math]\displaystyle{ Y\in \mathfrak{g} }[/math] and [math]\displaystyle{ K }[/math] is the Killing form. This defines a Lie bialgebra structure on [math]\displaystyle{ \mathfrak{g} }[/math], and is the "standard" example: it underlies the Drinfeld-Jimbo quantum group. Note that [math]\displaystyle{ \mathfrak{g'} }[/math] is solvable, whereas [math]\displaystyle{ \mathfrak{g} }[/math] is semisimple.
Relation to Poisson–Lie groups
The Lie algebra [math]\displaystyle{ \mathfrak{g} }[/math] of a Poisson–Lie group G has a natural structure of Lie bialgebra. In brief the Lie group structure gives the Lie bracket on [math]\displaystyle{ \mathfrak{g} }[/math] as usual, and the linearisation of the Poisson structure on G gives the Lie bracket on [math]\displaystyle{ \mathfrak{g^*} }[/math] (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 [math]\displaystyle{ f_1,f_2 \in C^\infty(G) }[/math] being two smooth functions on the group manifold. Let [math]\displaystyle{ \xi= (df)_e }[/math] be the differential at the identity element. Clearly, [math]\displaystyle{ \xi \in \mathfrak{g}^* }[/math]. The Poisson structure on the group then induces a bracket on [math]\displaystyle{ \mathfrak{g}^* }[/math], as
- [math]\displaystyle{ [\xi_1,\xi_2]=(d\{f_1,f_2\})_e\, }[/math]
where [math]\displaystyle{ \{,\} }[/math] is the Poisson bracket. Given [math]\displaystyle{ \eta }[/math] be the Poisson bivector on the manifold, define [math]\displaystyle{ \eta^R }[/math] to be the right-translate of the bivector to the identity element in G. Then one has that
- [math]\displaystyle{ \eta^R:G\to \mathfrak{g} \otimes \mathfrak{g} }[/math]
The cocommutator is then the tangent map:
- [math]\displaystyle{ \delta = T_e \eta^R\, }[/math]
so that
- [math]\displaystyle{ [\xi_1,\xi_2]= \delta^*(\xi_1 \otimes \xi_2) }[/math]
is the dual of the cocommutator.
See also
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. doi:10.1007/s00220-008-0578-2. Bibcode: 2009CMaPh.285..537B.
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