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.

MSC:

 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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