Lie bialgebra

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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.

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.

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

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}_{X}Y=[X,Y] is the adjoint.

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 \xi =(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 \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 {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.

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.
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