Primitive and Poisson Spectra of Twists of Polynomial Rings

Abstract

A family of flat deformations of a commutative polynomial ring S on n generators is considered, where each deformation B is a twist of S by a semisimple, linear automorphism σ of ℙ n−1, such that a Poisson bracket is induced on S. We show that if the symplectic leaves associated with this Poisson structure are algebraic, then they are the orbits of an algebraic group G determined by the Poisson bracket. In this case, we prove that if σ is 'generic enough', then there is a natural one-to-one correspondence between the primitive ideals of B and the symplectic leaves if and only if σ has a representative in GL(ℂ n ) which belongs to G. As an example, the results are applied to the coordinate ring $$\mathcal{O}_q (M_2 )$$ of quantum 2 × 2 matrices which is not a twist of a polynomial ring, although it is a flat deformation of one; if q is not a root of unity, then there is a bijection between the primitive ideals of $$\mathcal{O}_q (M_2 )$$ and the symplectic leaves.