Fundamentals of Poisson Lie Groups with Application to the Classical Double

Abstract

We give a constructive account of the fundamental ingredients of Poisson Lie theory as the basis for a description of the classical double group D. The double of a group G has a pointwise decomposition D ∼ G × G, where G and G are Lie subgroups generated by dual Lie algebras which form a Lie bialgebra. The double is an example of a factorisable Poisson Lie group, in the sense of Reshetikhin and Semenov-Tian-Shansky [1], and usually the study of its Poisson structures is developed only in the case when the subgroup G is itself factorisable. We give an explicit description of the Poisson Lie structure of the double without invoking this assumption. This is achieved by a direct calculation, in infinitesimal form, of the dressing actions of the subgroups on each other, and provides a new and general derivation of the Poisson Lie structure on the group G. For the example of the double of SU(2), the symplectic leaves of the Poisson Lie structures on SU(2) and SU(2) are displayed.

Cite this paper

@inproceedings{Ahluwalia1993FundamentalsOP, title={Fundamentals of Poisson Lie Groups with Application to the Classical Double}, author={K . S . Ahluwalia}, year={1993} }