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.

[edit] Definition

A Poisson-Lie group is a Lie group G for which the group multiplication \mu:G\times G\to G with μ(g1,g2) = g1g2 is a Poisson map, where the manifold G\times 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 f1 and f2 are real-valued, smooth functions on the Lie group, while g\, and g^\prime are elements of the Lie group. Here, Lg denotes left-multiplication and Rg denotes right-multiplication.

[edit] 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, it should be noted that neither left translations nor right translations are Poisson maps. Also, the inversion map \iota:G\to G taking ι(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 f1,f2 on G.

[edit] 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.
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