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" for which the group multiplication $mu:G\; imes\; G\; o\; G$ with $mu(g\_1,\; g\_2)=g\_1g\_2$ is a Poisson map, where the manifold $G\; imes\; G$ has been given the structure of a product Poisson manifold.

Explicitly, the following identity must hold for a Poisson–Lie group:

:$\{f\_1,f\_2\}\; (gg^prime)\; =\; \{f\_1\; circ\; L\_g,\; f\_2\; circ\; L\_g\}\; (g^prime)\; +\; \{f\_1\; circ\; R\_\{g^prime\},\; f\_2\; circ\; R\_\{g^prime\}\}\; (g)$

where $f\_1$ and $f\_2$ are real-valued, smooth functions on the Lie group, while $g,$ and $g^prime$ are elements of the Lie group. Here, $L\_g$ denotes left-multiplication and $R\_g$ denotes right-multiplication.

Homomorphisms

A Poisson–Lie group homomorphism $phi:G\; o\; H$ is defined to be both a Lie group homomorphism and a Poisson map. Although this is the "obvious" definition, it should be noted that neither left translations nor right translations are Poisson maps. Also, the inversion map $iota:G\; o\; G$ taking $iota(g)=g^\{-1\}$ is not a Poisson map either, although it is an anti-Poisson

:$\{f\_1\; circ\; iota,\; f\_2\; circ\; iota\; \}\; =\; -\{f\_1,\; f\_2\}\; circ\; iota$

for any two smooth functions $f\_1,\; f\_2$ on "G".

References

* 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.