Primitive and Poisson spectra of twists of polynomial rings. (English) Zbl 0939.16018

In this paper families of flat deformations of polynomial algebras \(S=\mathbb{C}[x_1,\dots,x_n]\) and of group algebras \(\mathbb{C}[x^{\pm 1}_1,\dots,x^{\pm 1}_n]\) are studied. The non-commutative deformed algebras are obtained by twisting the commutative multiplication using an automorphism \(\sigma\) of \({\mathcal P}^{n-1}\), so that a Poisson bracket is induced on \(S\). The focus of the paper examines when the primitive ideals of the deformed algebra are in bijection with the symplectic leaves associated with the Poisson structure on \(S\). An answer is obtained (for “generic” \(\sigma\)) in the case where the symplectic leaves are algebraic varieties. The work is motivated in part by the description of the prime spectrum of the quantized function algebras \({\mathcal O}_q(G)\) (for \(G\) semisimple and \(q\) generic) obtained by A. Joseph [J. Algebra 169, No. 2, 441-511 (1994; Zbl 0814.17013)], and so appropriately ends by applying the insights obtained to the prime spectrum of \({\mathcal O}_q(M_2(\mathbb{C}))\), the coordinate ring of quantum \(2\times 2\) matrices.


16S80 Deformations of associative rings
17B37 Quantum groups (quantized enveloping algebras) and related deformations
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
16S36 Ordinary and skew polynomial rings and semigroup rings
14A22 Noncommutative algebraic geometry
14E07 Birational automorphisms, Cremona group and generalizations


Zbl 0814.17013
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