Semiclassical Limits of Quantum Affine Spaces

Details Semiclassical Limits of Quantum Affine Spaces Semiclassical Limits of Quantum Affine Spaces

2007-08-08

arxiv.org/abs/0708.1091v2

Mathematics

Quantum Algebra

20 pages; LaTeX; Xy-pic; 4 diagrams; to appear in Proc. Edinburgh Math. Soc

Scientific paper

Semiclassical limits of generic multiparameter quantized coordinate rings A = O_q(k^n) of affine spaces are constructed and related to A, for k an algebraically closed field of characteristic zero and q a multiplicatively antisymmetric matrix whose entries generate a torsionfree subgroup of k*. A semiclassical limit of A is a Poisson algebra structure on the corresponding classical coordinate ring R = O(k^n), and results of Oh, Park, Shin and the authors are used to construct homeomorphisms from the Poisson prime and Poisson primitive spectra of R onto the prime and primitive spectra of A. The Poisson primitive spectrum of R is then identified with the space of symplectic cores in k^n in the sense of Brown and Gordon, and an example is presented (over the complex numbers) for which the Poisson primitive spectrum of R is not homeomorphic to the space of symplectic leaves in k^n. Finally, these results are extended from quantum affine spaces to quantum affine toric varieties.

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