Lie bialgebra

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

A vector space ${\mathfrak {g}}$ is a Lie bialgebra if it is a Lie algebra, and there is the structure of Lie algebra also on the dual vector space $\mathfrak{g}^*$ which is compatible. More precisely the Lie algebra structure on ${\mathfrak {g}}$ is given by a Lie bracket $[\ ,\ ]:{\mathfrak {g}}\otimes {\mathfrak {g}}\to {\mathfrak {g}}$ and the Lie algebra structure on $\mathfrak{g}^*$ is given by a Lie bracket $\delta ^{*}:{\mathfrak {g}}^{*}\otimes {\mathfrak {g}}^{*}\to {\mathfrak {g}}^{*}$ . Then the map dual to $\delta^*$ is called the cocommutator, $\delta :{\mathfrak {g}}\to {\mathfrak {g}}\otimes {\mathfrak {g}}$ and the compatibility condition is the following cocyle relation:

\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. Note that this definition is symmetric and $\mathfrak{g}^*$ is also a Lie bialgebra, the dual Lie bialgebra.

Example

Let ${\mathfrak {g}}$ 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 ${\mathfrak {t}}\subset {\mathfrak {g}}$ and a choice of positive roots. Let ${\mathfrak {b}}_{\pm }\subset {\mathfrak {g}}$ be the corresponding opposite Borel subalgebras, so that ${\mathfrak {t}}={\mathfrak {b}}_{-}\cap {\mathfrak {b}}_{+}$ and there is a natural projection $\pi :{\mathfrak {b}}_{\pm }\to {\mathfrak {t}}$ . Then define a Lie algebra

{\mathfrak {g'}}:=\{(X_{-},X_{+})\in {\mathfrak {b}}_{-}\times {\mathfrak {b}}_{+}\ {\bigl \vert }\ \pi (X_{-})+\pi (X_{+})=0\}

which is a subalgebra of the product ${\mathfrak {b}}_{-}\times {\mathfrak {b}}_{+}$ , and has the same dimension as ${\mathfrak {g}}$ . Now identify ${\mathfrak {g'}}$ with dual of ${\mathfrak {g}}$ via the pairing

\langle (X_{-},X_{+}),Y\rangle :=K(X_{+}-X_{-},Y)

where $Y\in {\mathfrak {g}}$ and $K$ is the Killing form. This defines a Lie bialgebra structure on ${\mathfrak {g}}$ , and is the "standard" example: it underlies the Drinfeld-Jimbo quantum group. Note that ${\mathfrak {g'}}$ is solvable, whereas ${\mathfrak {g}}$ is semisimple.

Relation to Poisson-Lie groups

The Lie algebra ${\mathfrak {g}}$ of a Poisson-Lie group G has a natural structure of Lie bialgebra. In brief the Lie group structure gives the Lie bracket on ${\mathfrak {g}}$ as usual, and the linearisation of the Poisson structure on G gives the Lie bracket on ${\mathfrak {g^{*}}}$ (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, $\xi \in {\mathfrak {g}}^{*}$ . 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.

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.
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. doi:10.1007/s00220-008-0578-2.

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Lie bialgebra

Definition

Example

Relation to Poisson-Lie groups

See also

References