Family of Poisson Structures on Compact Hermitian Symmetric Spaces

摘要

The work is the first part of a program devoted to obtaining a betterunderstanding of the interrelations between the geometry of a Poisson manifold and both classical and quantum symmetries of field theoretical models. By a well-known prescription of Kirillov-Kostant-Souriau coadjoint orbits O of a compact semisimple Lie group G have a natural symplectic structure and hence are Poisson manifolds. The developments of the Poisson-Lie group theory make it possible to introduce another Poisson structure on O which is induced from the Poisson-Lie structure on the compact semi-simple Lie group G. The authors denote these structures by pi(sub Kir) and pi(sub G) respectively. The main result of the paper is to show that pi(sub Kir) and pi(sub G) are coordinated (or compatible) on the orbit O iff O is a hermitian compact symmetric space occuring in the Cartan list.