Poisson–Lie group

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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.


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_1g_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, Lg denotes left-multiplication and Rg 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.


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.


  • 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.