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 ${\displaystyle \mu :G\times G\to G}$ with ${\displaystyle \mu (g_{1},g_{2})=g_{1}g_{2}}$ 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:

${\displaystyle \{f_{1},f_{2}\}(gg')=\{f_{1}\circ L_{g},f_{2}\circ L_{g}\}(g')+\{f_{1}\circ R_{g^{\prime }},f_{2}\circ R_{g'}\}(g)}$

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

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

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

Homomorphisms

A Poisson–Lie group homomorphism ${\displaystyle \phi :G\to H}$ 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 ${\displaystyle \iota :G\to G}$ taking ${\displaystyle \iota (g)=g^{-1}}$ is not a Poisson map either, although it is an anti-Poisson map:

${\displaystyle \{f_{1}\circ \iota ,f_{2}\circ \iota \}=-\{f_{1},f_{2}\}\circ \iota }$

for any two smooth functions ${\displaystyle f_{1},f_{2}}$ 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 Ltd. ISBN 0-521-55884-0.