# 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 algebra of a Poisson–Lie group is a Lie bialgebra.

## 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×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 f1 and f2 are real-valued, smooth functions on the Lie group, while g and g' are elements of the Lie group. Here, Lg denotes left-multiplication and Rg denotes right-multiplication.

If ${\mathcal {P}}$  denotes the corresponding Poisson bivector on G, the condition above can be equivalently stated as

${\mathcal {P}}(gg')=L_{g\ast }({\mathcal {P}}(g'))+R_{g'\ast }({\mathcal {P}}(g))$

Note that for Poisson-Lie group always $\{f,g\}(e)=0$ , or equivalently ${\mathcal {P}}(e)=0$ . This means that non-trivial Poisson-Lie structure is never symplectic, not even of constant rank.

## 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.