Graded automorphisms of quantum affine spaces
Hai Jin
TL;DR
The paper addresses the problem of classifying the graded automorphism group $Aut_{gr}(\\mathcal{O}_{\\bf q}(k^n))$ for arbitrary multi-parameter quantum affine spaces, removing prior restrictions on the parameter matrix $\\bf q$. It develops a criterion characterizing when a matrix $m$ induces a graded automorphism and introduces a block- and permutation-based framework, including skeleton and compatible permutations and a block decomposition, to capture the symmetries of the quantum relations. The main result is a precise description: $Aut_{gr}(\\mathcal{O}_{\\bf q}(k^n))\\cong (\\prod_{i=1}^m GL_{|B_i|}(k)) \\rtimes (\\mathcal P_{\\bf q}/\\mathcal I_{\\bf q})$, providing a unifying semidirect-product structure that generalizes known cases and enables computation for broader quantum spaces. This advances the understanding of automorphism groups in noncommutative algebra and has potential implications for related areas such as quantum clusters and noncommutative geometry.
Abstract
This paper computes the graded automorphism group of quantum affine spaces. Specifically, we determine that this group is isomorphic to a semi-direct product of a blocked diagonal matrix group and a permutation group.