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Semiclassical limits of quantized coordinate rings. (English) Zbl 1202.16027

Dinh Van Huynh (ed.) et al., Advances in ring theory. Papers of the conference on algebra and applications, Athens, OH, USA, June 18–21, 2008. Basel: Birkhäuser (ISBN 978-3-0346-0285-3/hbk). Trends in Mathematics, 165-204 (2010).
Let \(A\) be the coordinate algebra \(\mathcal O(V)\) of some quantum space or group with parameter of quantization \(q\). A semiclassical limit of \(A\) is an algebra over the Laurent polynomial algebra \(k[t^\pm]\) where \(q\) in the defining relations of \(A\) is replaced by \(t\). The paper considers connections between the prime spectrum of \(A\) and symplectic leaves of the semiclassical limit, between primitive ideals in \(A=\mathcal O(V)\) and the symplectic core in \(V\). Some conjectures are formulated.
For the entire collection see [Zbl 1184.13004].

MSC:

16T20 Ring-theoretic aspects of quantum groups
16S38 Rings arising from noncommutative algebraic geometry
17B37 Quantum groups (quantized enveloping algebras) and related deformations
17B63 Poisson algebras
16D25 Ideals in associative algebras
53D17 Poisson manifolds; Poisson groupoids and algebroids
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