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

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A vector space   is a Lie bialgebra if it is a Lie algebra, and there is the structure of Lie algebra also on the dual vector space   which is compatible. More precisely the Lie algebra structure on   is given by a Lie bracket   and the Lie algebra structure on   is given by a Lie bracket  . Then the map dual to   is called the cocommutator,   and the compatibility condition is the following cocycle relation:

 

where   is the adjoint. Note that this definition is symmetric and   is also a Lie bialgebra, the dual Lie bialgebra.

Example

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Let   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   and a choice of positive roots. Let   be the corresponding opposite Borel subalgebras, so that   and there is a natural projection  . Then define a Lie algebra

 

which is a subalgebra of the product  , and has the same dimension as  . Now identify   with dual of   via the pairing

 

where   and   is the Killing form. This defines a Lie bialgebra structure on  , and is the "standard" example: it underlies the Drinfeld-Jimbo quantum group. Note that   is solvable, whereas   is semisimple.

Relation to Poisson–Lie groups

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The Lie algebra   of a Poisson–Lie group G has a natural structure of Lie bialgebra. In brief the Lie group structure gives the Lie bracket on   as usual, and the linearisation of the Poisson structure on G gives the Lie bracket on   (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   being two smooth functions on the group manifold. Let   be the differential at the identity element. Clearly,  . The Poisson structure on the group then induces a bracket on  , as

 

where   is the Poisson bracket. Given   be the Poisson bivector on the manifold, define   to be the right-translate of the bivector to the identity element in G. Then one has that

 

The cocommutator is then the tangent map:

 

so that

 

is the dual of the cocommutator.

See also

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References

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  • 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. arXiv:0708.1762. Bibcode:2009CMaPh.285..537B. doi:10.1007/s00220-008-0578-2. S2CID 8946457.