Semiclassical limits of quantum affine spaces. (English) Zbl 1184.16037

An affine multiparameter quantum space \(A\) over an algebraically closed field \(k\) of characteristic zero is generated by elements \(x_1,\dots,x_n\) with defining relations \(x_ix_j=q_{ij}x_jx_i\) where \(q_{ij}\in k^*\). It is assumed that the multiplicative subgroup \(G\) in \(k^*\) generated by all \(q_{ij}\) is torsion-free. Let \(R=k[z_1,\dots,z_n]\) be the ordinary polynomial algebra. There is defined a Poisson bracket \(\{z_i,z_j\}=q_{ij}'z_iz_j\) in \(R\) for some \(q_{ij}'\in G\).
The main result of the paper establishes a homeomorphism between the Poisson-prime spectrum of \(R\) and the prime spectrum of \(A\). This homeomorphism maps Poisson-primitive ideals of \(R\) onto primitive ideals of \(A\).


16T20 Ring-theoretic aspects of quantum groups
16D25 Ideals in associative algebras
16S38 Rings arising from noncommutative algebraic geometry
16S36 Ordinary and skew polynomial rings and semigroup rings
17B37 Quantum groups (quantized enveloping algebras) and related deformations
53D17 Poisson manifolds; Poisson groupoids and algebroids
Full Text: DOI arXiv