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Semiclassical Limits of Quantized Coordinate Rings

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Book cover Advances in Ring Theory

Part of the book series: Trends in Mathematics ((TM))

Abstract

This paper offers an expository account of some ideas, methods, and conjectures concerning quantized coordinate rings and their semiclassical limits, with a particular focus on primitive ideal spaces. The semiclassical limit of a family of quantized coordinate rings of an affine algebraic variety V consists of the classical coordinate ring O(V) equipped with an associated Poisson structure. Conjectured relationships between primitive ideals of a generic quantized coordinate ring A and symplectic leaves in V (relative to a semiclassical limit Poisson structure on O(V)) are discussed, as are breakdowns in the connections when the symplectic leaves are not algebraic. This prompts replacement of the differential-geometric concept of symplectic leaves with the algebraic concept of symplectic cores, and a reformulated conjecture is proposed: The primitive spectrum of A should be homeomorphic to the space of symplectic cores in V, and to the Poisson-primitive spectrum of O(V). Various examples, including both quantized coordinate rings and enveloping algebras of solvable Lie algebras, are analyzed to support the choice of symplectic cores to replace symplectic leaves.

This research was partially supported by National Science Foundation grant DMS-0800948.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Santa Barbara, CA, 93106, USA

    K. R. Goodearl

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  1. K. R. Goodearl
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  1. Department of Mathematics, Ohio University, 321 Morton Hall, Athens, OH, 45701, USA

    Dinh Van Huynh  & Sergio R. López-Permouth  & 

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Dedicated to S.K. Jain on the occasion of his 70th birthday

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Goodearl, K.R. (2010). Semiclassical Limits of Quantized Coordinate Rings. In: Van Huynh, D., López-Permouth, S.R. (eds) Advances in Ring Theory. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0286-0_12

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