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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.
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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.  
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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 equation]]s.
 
==Definition==
More precisely, comultiplication on the algebra, <math>\delta:\mathfrak{g} \to \mathfrak{g} \otimes \mathfrak{g}</math>, is called the '''cocommutator''', and must satisfy two properties. The dual
 
:<math>\delta^*:\mathfrak{g}^* \otimes \mathfrak{g}^* \to \mathfrak{g}^*</math>
must be a Lie bracket on <math>\mathfrak{g}^*</math>, and it must be a cocycle:
 
:<math>\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>\operatorname{ad}_XY=[X,Y]</math> is the adjoint.
 
==Relation to Poisson-Lie groups==
Let ''G'' be a Poisson-Lie group, with <math>f_1,f_2 \in C^\infty(G)</math> being two smooth functions on the group manifold. Let <math>\xi= (df)_e</math> be the differential at the identity element. Clearly, <math>\xi \in \mathfrak{g}^*</math>. The [[Poisson structure]] on the group then induces a bracket on <math>\mathfrak{g}^*</math>, as
 
:<math>[\xi_1,\xi_2]=(d\{f_1,f_2\})_e\,</math>
 
where <math>\{,\}</math> is the [[Poisson bracket]]. Given <math>\eta</math> be the [[Poisson bivector]] on the manifold, define <math>\eta^R</math> to be the right-translate of the bivector to the identity element in ''G''.  Then one has that
 
:<math>\eta^R:G\to \mathfrak{g} \otimes \mathfrak{g}</math>
 
The cocommutator is then the tangent map:
 
:<math>\delta = T_e \eta^R\,</math>
 
so that
 
:<math>[\xi_1,\xi_2]= \delta^*(\xi_1 \otimes \xi_2)</math>
 
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.
 
[[Category:Lie algebras]]
[[Category:Coalgebras]]
[[Category:Symplectic geometry]]

Revision as of 03:49, 4 March 2014

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