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In mathematics, a Lie bialgebra is the Lie-theoretic case of a
bialgebra In mathematics, a bialgebra over a field ''K'' is a vector space over ''K'' which is both a unital associative algebra and a counital coassociative coalgebra. The algebraic and coalgebraic structures are made compatible with a few more axioms. ...
: it is a set with a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
and a Lie coalgebra structure which are compatible. It is a
bialgebra In mathematics, a bialgebra over a field ''K'' is a vector space over ''K'' which is both a unital associative algebra and a counital coassociative coalgebra. The algebraic and coalgebraic structures are made compatible with a few more axioms. ...
where the multiplication is skew-symmetric and satisfies a dual
Jacobi identity In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the asso ...
, so that the dual vector space is a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
, 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 In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
of a Poisson–Lie group. Lie bialgebras occur naturally in the study of the
Yang–Baxter equation In physics, the Yang–Baxter equation (or star–triangle relation) is a consistency equation which was first introduced in the field of statistical mechanics. It depends on the idea that in some scattering situations, particles may preserve thei ...
s.


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 In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. T ...
. Given \eta be the
Poisson bivector In differential geometry, a Poisson structure on a smooth manifold M is a Lie bracket \ (called a Poisson bracket in this special case) on the algebra (M) of smooth functions on M , subject to the Leibniz rule : \ = \h + g \ . Equivalentl ...
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 Lie algebras Coalgebras Symplectic geometry