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Primitive ideals of the coordinate ring of quantum symplectic space. (English) Zbl 0828.17011
The author classifies the primitive spectrum of the coordinate ring \(R:= {\mathcal O}_q ({\mathfrak {sp}} \mathbb{C}^{2n})\) of quantum symplectic space when \(q\) is not a root of unity. This coordinate ring turns out to be an iterated Ore extension over \(\mathbb{C}\), so the theory of skew polynomial rings plays a major role in the classification. The author replaces the canonical set of generators of \(R\) by a larger set \(P\), defines a notion of admissible subset of \(P\), and then uses admissible subsets to parametrize the primitive spectrum of \(R\). Some features of enveloping algebras of solvable Lie algebras reappear in this setting; all prime ideals are completely prime, all primitive factor algebras have even Gelfand-Kirillov dimension, and there is an analogue of the Tauvel height formula. On the other hand, the center of \(R\) reduces to \(\mathbb{C}\).

MSC:
17B37 Quantum groups (quantized enveloping algebras) and related deformations
16S30 Universal enveloping algebras of Lie algebras
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