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, 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

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

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.

See also

• Lie bialgebra
• Quantum group
• Affine quantum group
• Quantum affine algebras

References

• Doebner, H.-D.; Hennig, J.-D., eds. (1989). Quantum groups. Proceedings of the 8th International Workshop on Mathematical Physics, Arnold Sommerfeld Institute, Claausthal, FRG. Berlin: Springer-Verlag. ISBN 3-540-53503-9.
• Chari, Vyjayanthi; Pressley, Andrew (1994). A Guide to Quantum Groups. Cambridge: Cambridge University Press. ISBN 0-521-55884-0.