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.

Definition[edit]

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 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 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[edit]

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.

See also[edit]

References[edit]

  • 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.