Vanishing of derivations and prime spectra of quantum algebras
The object of study in the paper is an iterated Ore extension \[ k=A_0\subset k[X_1]=A_1\subset k[X_1][X_2;\sigma_2,\delta_2]=A_2\subset \cdots\subset k[X_1][X_2;\sigma_2,\delta_2]\cdots[X_N;\sigma_N,\delta_N]=A_N,\quad N\geq 2, \] where \(k\) is a field, \(\sigma_j\) is an automorphism and \(\delta_j\) is a locally nilpotent \(\sigma_j\)-derivation of \(A_{j-1}\). Moreover \(\sigma_j\delta_j=q_j\delta_j\sigma_j\) for some \(q_j\in k^*\), \(j\geq 2\), which is not a root of 1 and \(\sigma_j(X_i)=\lambda_{ji}X_i\), \(i<j\), where \(\lambda_{ji}\in k^*\). The algebra \(R=A_N\) is an Ore domain with a division ring of fractions \(F=\text{Fract }R\). The paper presents an algorithm for search of a new system of variables \(T_1,\ldots,T_N\) in \(F\) generating a quantum affine space \(\overline R\subset F\) with some nice properties. For example there is a natural embedding \(\phi\colon\text{Spec}\;R\to\text{Spec}\;\overline R\) such that \(\text{Fract}(R/P)\simeq\text{Fract}(\overline R/\phi(P))\) for any \(P\in\text{Spec}\;R\). There is given some additional information concerning the map \(\phi\). For an application the author considers the cases when \(R\) is a quantum universal enveloping algebra, coordinate algebra of a quantum group, quantum Weyl algebra, quantum Euclidean space etc.
scientific article; zbMATH DE number 1901834
scientific article