Lie bialgebra

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 $[\backslash \; ,\backslash \; ]:\backslash mathfrak\{g\}\; \backslash otimes\; \backslash mathfrak\{g\}\; \backslash to\; \backslash mathfrak\{g\}$

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.

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.

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.

• 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, {{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}}.
• {{cite journal | last1 = Beisert | first1 = N. | last2 = Spill | first2 = F. | year = 2009 | title = The classical r-matrix of AdS/CFT and its Lie bialgebra structure | journal = Communications in Mathematical Physics | volume = 285 | issue = 2| pages = 537–565 | doi = 10.1007/s00220-008-0578-2 | arxiv = 0708.1762 | bibcode = 2009CMaPh.285..537B | s2cid = 8946457 }}

Category:Lie algebras

Category:Coalgebras

Category:Symplectic geometry