Primitive and Poisson Spectra of Twists of Polynomial Rings |
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Authors: | Michaela Vancliff |
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Affiliation: | (1) Department of Mathematics, University of Texas at Arlington, Box 19408, Arlington, TX, 76019-0408, U.S.A. |
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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 of 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 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 has a representative in GL(n) which belongs to G. As an example, the results are applied to the coordinate ring 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 and the symplectic leaves. |
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Keywords: | twisted homogeneous coordinate ring symplectic leaf Poisson manifold primitive ideal quantum matrices |
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