Triviality of the automorphism group of the multiparameter quantum affine $n$-space

Abstract

A multiparameter quantum affine space of rank $n$ is the $\mathbb F$-algebra generated by indeterminates $X_1, \cdots, X_n$ satisfying $X_iX_j = q_{ij} X_jX_i \ (1 \le i < j \le n)$ where $q_{ij}$ are nonzero scalars in $\mathbb F^\ast$. The corresponding quantum torus is generated by the $X_i$ and together with their inverses subject to the same relations. So far the automorphisms of a quantum affine space have been considered mainly in the uniparameter case, that is, $q_{ij} = q$. We remove this restriction here. Necessary and sufficient conditions are obtained for the quantum affine space to be rigid, that is, the only automorphisms are the trivial ones arising from the action of the torus $(\mathbb F^\ast)^n$. These conditions are based on the multiparameters $q_{ij}$ and also on the subgroup of $\mathbb F^\ast$ generated by these multiparameters. We employ the results in J. Alev and M. Chamarie, Derivations et automorphismes de quelques algebras quantiques, Communications in Algebra, 1992 (20), 1787-1802, and point out a small error in a main theorem in this paper which however remains valid with a small modification. We also note that a quantum affine space whose corresponding quantum torus has dimension one necessarily has a trivial automorphism group. This is a consequence of a result of J.~M.~Osborne, D.~S.~Passman, Derivations of Skew Polynomial Rings, J. Algebra, 1995, 176, 417--448. We expand the known list of examples of quantum tori that have dimension one and are thus hereditary noetherian domains.