Lie bialgebra

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In mathematics, a Lie bialgebra is the Lie-theoretic case of a bialgebra: its a set with a Lie algebra and a Lie coalgebra structure which are compatible.

It is a bialgebra where the comultiplication is skew-symmetric, so that its dual is a Lie bracket, and such that the comultiplication is a 1-cocycle (i.e. dual to the Jacobi identity). 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.

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[edit] Definition

More precisely, comultiplication on the algebra, \delta:\mathfrak{g} \to \mathfrak{g} \otimes \mathfrak{g}, is called the cocommutator, and must satisfy two properties. The dual

\delta^*:\mathfrak{g}^* \otimes \mathfrak{g}^* \to \mathfrak{g}^*

must be a Lie bracket on \mathfrak{g}^*, and it must be a cocycle:

\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.

[edit] Relation to Poisson-Lie groups

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 ξ = (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 η be the Poisson bivector on the manifold, define ηR 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.

[edit] See also


[edit] 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.
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