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Fundamentals of Poisson Lie Groups with Application to the Classical Double | | K. S. Ahluwalia
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13 Oct 1993 | | Subject: | High Energy Physics - Theory; Quantum Algebra | hep-th math.QA | | 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 $Dsim G imes G^*$, 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. | | Source: | arXiv, hep-th/9310068 | | Services: | Forum | Review | PDF | Favorites | |
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