On trigonometric Calogero-Moser spaces

In this paper, we investigate the trigonometric Calogero-Moser space C 3 tri as an irreducible affine variety and analyze the detailed structure of its coordinate ring C [ C 3 tri ] . We establish and optimize the Poisson structure on C 3 tri , deriving the Poisson bracket multiplication table for the generators a 1 , … , a 9 . Furthermore, we identify five defining relations r 1 , … , r 5 among the generators and establish the isomorphism C [ C 3 tri ] ≅ C [ a 1 , … , a 9 ] / I , where I is the ideal generated by the five relations.