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DEFINITION

A Poisson-Lie group is a Lie group ''G'' for which the group multiplication \mu:G imes G o G with \mu(g_1, g_2)=g_1g_2 is a Poisson Map , where the manifold G imes 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 f_1 and f_2 are real-valued, smooth functions on the Lie group, while g\, and g^\prime are elements of the Lie group. Here, L_g denotes left-multiplication and R_g denotes right-multiplication.


HOMOMORPHISMS

A Poisson-Lie group homomorphism \phi:G o 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 o 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''.


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.