an:01385253
Zbl 0939.16018
Vancliff, Michaela
Primitive and Poisson spectra of twists of polynomial rings
EN
Algebr. Represent. Theory 2, No. 3, 269-285 (1999).
00059736
1999
j
16S80 17B37 16W10 16S36 14A22 14E07
twisted homogeneous coordinate rings; Poisson manifolds; flat deformations; polynomial algebras; group algebras; Poisson brackets; primitive ideals; deformed algebras; symplectic leaves; prime spectra; quantized function algebras; coordinate rings; quantum \(2\times 2\) matrices
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 \textit{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.
K.A.Brown (Glasgow)
Zbl 0814.17013