# 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 with 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:

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*denotes right-multiplication.

_{g}If denotes the corresponding Poisson bivector on *G*, the condition above can be equivalently stated as

Note that for Poisson-Lie group always , or equivalently . This means that non-trivial Poisson-Lie structure is never symplectic, not even of constant rank.

## Homomorphisms

A Poisson–Lie group homomorphism 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 taking is not a Poisson map either, although it is an anti-Poisson map:

for any two smooth functions on *G*.

## References

- 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 - 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