Lie bialgebra

 ( Lie Algebras ) 

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.

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

$\backslash delta(X,Y)\; =\; \backslash left(\backslash operatorname\{ad\}\_X\; \backslash otimes\; 1\; +\; 1\; \backslash otimes\; \backslash operatorname\{ad\}\_X\backslash right)\; \backslash delta(Y)\; -\; \backslash left(\backslash operatorname\{ad\}\_Y\; \backslash otimes\; 1\; +\; 1\; \backslash otimes\; \backslash operatorname\{ad\}\_Y\backslash right)\; \backslash delta(X)$

where $\backslash operatorname\{ad\}\_XY=X,Y$ is the adjoint. Note that this definition is symmetric and $\backslash mathfrak\{g\}^*$ is also a Lie bialgebra, the dual Lie bialgebra.

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

$\backslash mathfrak\{g\text{'}\}\; :=\; \backslash \{\; (X\_-,X\_+)\backslash in\; \backslash mathfrak\{b\}\_-\; \backslash times\; \backslash mathfrak\{b\}\_+\backslash \; \backslash bigl\backslash vert\backslash \; \backslash pi(X\_-)\; +\; \backslash pi(X\_+)\; =\; 0\backslash \}$

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

$\backslash langle\; (X\_-,X\_+),\; Y\; \backslash rangle\; :=\; K(X\_+\; -\; X\_-,\; Y)$

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

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Relation to Poisson–Lie groups The Lie algebra $\backslash mathfrak\{g\}$ of a Poisson–Lie group G has a natural structure of Lie bialgebra. In brief the Lie group structure gives the Lie bracket on $\backslash mathfrak\{g\}$ as usual, and the linearisation of the Poisson structure on G gives the Lie bracket on $\backslash mathfrak\{g^*\}$ (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 $f\_1,f\_2\; \backslash in\; C^\backslash infty(G)$ being two smooth functions on the group manifold. Let $\backslash xi=\; (df)\_e$ be the differential at the identity element. Clearly, $\backslash xi\; \backslash in\; \backslash mathfrak\{g\}^*$. The Poisson structure on the group then induces a bracket on $\backslash mathfrak\{g\}^*$, as

$\backslash xi\_1,\backslash xi\_2\; =\; (d\backslash \{f\_1,f\_2\backslash \})\_e\backslash ,$

where $\backslash \{,\backslash \}$ is the Poisson bracket. Given $\backslash eta$ be the Poisson bivector on the manifold, define $\backslash eta^R$ to be the right-translate of the bivector to the identity element in G. Then one has that

$\backslash eta^R:G\backslash to\; \backslash mathfrak\{g\}\; \backslash otimes\; \backslash mathfrak\{g\}$

The cocommutator is then the tangent map:

$\backslash delta\; =\; T\_e\; \backslash eta^R\backslash ,$

so that

$\backslash xi\_1,\backslash xi\_2=\; \backslash delta^*(\backslash xi\_1\; \backslash otimes\; \backslash xi\_2)$

is the dual of the cocommutator.

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See also

• Lie coalgebra
• Manin triple

• 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, .
• Vyjayanthi Chari and Andrew Pressley, A Guide to Quantum Groups, (1994), Cambridge University Press, Cambridge .

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