# Poisson–Lie group

In mathematics, a Poisson–Lie group is a Poisson manifold that is also a Lie group, with the group multiplication being compatible with the Poisson algebra structure on the manifold.
The infinitesimal counterpart of a Poisson–Lie group is a Lie bialgebra, in analogy to Lie algebras as the infinitesimal counterparts of Lie groups.
Many quantum groups are quantizations of the Poisson algebra of functions on a Poisson–Lie group.

## Definition

A Poisson–Lie group is a Lie group $G$ equipped with a Poisson bracket for which the group multiplication $\mu :G\times G\to G$ with $\mu (g_{1},g_{2})=g_{1}g_{2}$ is a Poisson map, where the manifold $G\times G$ has been given the structure of a product Poisson manifold.
Explicitly, the following identity must hold for a Poisson–Lie group:
    $\{f_{1},f_{2}\}(gg')=\{f_{1}\circ L_{g},f_{2}\circ L_{g}\}(g')+\{f_{1}\circ R_{g^{\prime }},f_{2}\circ R_{g'}\}(g)$
where $f_{1}$ and $f_{2}$ are real-valued, smooth functions on the Lie group, while $g$ and $g'$ are elements of the Lie group. Here, $L_{g}$ denotes left-multiplication and $R_{g}$ denotes right-multiplication.
If ${\mathcal {P}}$ denotes the corresponding Poisson bivector on $G$, the condition above can be equivalently stated as
    ${\mathcal {P}}(gg')=L_{g\ast }({\mathcal {P}}(g'))+R_{g'\ast }({\mathcal {P}}(g))$
In particular, taking $g=g'=e$ one obtains ${\mathcal {P}}(e)=0$, or equivalently $\{f,g\}(e)=0$. Applying Weinstein splitting theorem to $e$ one sees that non-trivial Poisson-Lie structure is never symplectic, not even of constant rank.

## Poisson-Lie groups - Lie bialgebra correspondence

The Lie algebra ${\mathfrak {g}}$ of a Poisson–Lie group has a natural structure of Lie coalgebra given by linearising the Poisson tensor $P:G\to TG\wedge TG$ at the identity, i.e. δ:=d$_e$P:g→g∧g{\textstyle \delta :=d_{e}P:{\mathfrak {g}}\to {\mathfrak {g}}\wedge {\mathfrak {g}}} is a comultiplication. Moreover, the algebra and the coalgebra structure are compatible, i.e. ${\mathfrak {g}}$ is a Lie bialgebra,
The classical Lie group–Lie algebra correspondence, which gives an equivalence of categories between simply connected Lie groups and finite-dimensional Lie algebras, was extended by Drinfeld to an equivalence of categories between simply connected Poisson–Lie groups and finite-dimensional Lie bialgebras.
Thanks to Drinfeld theorem, any Poisson–Lie group $G$ has a dual Poisson–Lie group, defined as the Poisson–Lie group integrating the dual ${\mathfrak {g}}^{*}$ of its bialgebra.

## Homomorphisms

A Poisson–Lie group homomorphism $\phi :G\to H$ is defined to be both a Lie group homomorphism and a Poisson map. Although this is the "obvious" definition, neither left translations nor right translations are Poisson maps. Also, the inversion map $\iota :G\to G$ taking $\iota (g)=g^{-1}$ is not a Poisson map either, although it is an anti-Poisson map:
    $\{f_{1}\circ \iota ,f_{2}\circ \iota \}=-\{f_{1},f_{2}\}\circ \iota$
for any two smooth functions $f_{1},f_{2}$ on $G$.

## Examples
### Trivial examples

* Any trivial Poisson structure on a Lie group $G$ defines a Poisson–Lie group structure, whose bialgebra is simply ${\mathfrak {g}}$ with the trivial comultiplication.
* The dual ${\mathfrak {g}}^{*}$ of a Lie algebra, together with its linear Poisson structure, is an additive Poisson–Lie group.
These two example are dual of each other via Drinfeld theorem, in the sense explained above.

### Other examples

Let $G$ be any semisimple Lie group. Choose a maximal torus $T\subset G$ and a choice of positive roots. Let $B_{\pm }\subset G$ be the corresponding opposite Borel subgroups, so that $T=B_{-}\cap B_{+}$ and there is a natural projection $\pi :B_{\pm }\to T$. Then define a Lie group
    $G^{*}:=\{(g_{-},g_{+})\in B_{-}\times B_{+}\ {\bigl \vert }\ \pi (g_{-})\pi (g_{+})=1\}$
which is a subgroup of the product $B_{-}\times B_{+}$, and has the same dimension as $G$.
The standard Poisson–Lie group structure on $G$ is determined by identifying the Lie algebra of $G^{*}$ with the dual of the Lie algebra of $G$, as in the standard Lie bialgebra example. This defines a Poisson–Lie group structure on both $G$ and on the dual Poisson Lie group $G^{*}$. This is the "standard" example: the Drinfeld-Jimbo quantum group $U_{q}{\mathfrak {g}}$ is a quantization of the Poisson algebra of functions on the group $G^{*}$. Note that $G^{*}$ is solvable, whereas $G$ is semisimple.