Double-clad fiber

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:

$\{f_1,f_2\}(g{g}^{\prime })=\{f_1\circ L_g,f_2\circ L_g\}(g^{\prime })+\{f_1\circ R_{g^{\prime }},f_2\circ R_{g^{\prime }}\}(g)$

where f₁ and f₂ are real-valued, smooth functions on the Lie group, while g and g' are elements of the Lie group. Here, 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

$\mathcal{𝒫}(g{g}^{\prime })=L_{g\ast }(\mathcal{𝒫}(g^{\prime }))+R_{g^{\prime }\ast }(\mathcal{𝒫}(g))$

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 $\varphi :G\to H$ 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.

References

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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534