Stabilizing Automorphisms of Quantum Affine Space
Ethan Jensen, Anne Shepler
TL;DR
This work determines the graded automorphism groups $Aut_{gr}(S_q(V))$ of quantum affine spaces for $\dim V \le 7$. It develops a semidirect-product framework $Aut_{gr}(S_q(V)) \cong (\prod_{B\in \mathcal{B}_q} GL(V_B)) \rtimes Stab(q)$ and a block-structure approach to decompose and construct these groups, including direct-product and Kronecker-product techniques. The authors establish infinite families of groups realizable as $Aut_{gr}(S_q(V))$, provide a practical $O(n^2)$ partition-based method to compute decompositions, and deliver a complete low-dimensional classification, distinguishing monomial from nonmonomial cases and detailing required field sizes. The results yield a comprehensive landscape of Aut$_{gr}$ for quantum polynomial rings up to dimension seven and offer a computational framework for exploring automorphism structures in noncommutative coordinate rings.
Abstract
We examine the graded automorphism groups of quantum affine spaces and classify these groups for spaces of dimension 7 or less. Using permutation actions on partitions, we investigate cases when the group decomposes as a product of graded automorphism groups of smaller dimensional spaces, and we describe the groups arising from the Kronecker tensor product of independent quantum parameter matrices.