Complete Structural Analysis of $q$-Heisenberg Algebras: Homology, Rigidity, Automorphisms, and Deformations

Abstract

This paper establishes several fundamental structural properties of the $q$-Heisenberg algebra $\mathfrak{h}_n(q)$, a quantum deformation of the classical Heisenberg algebra. We first prove that when $q$ is not a root of unity, the global homological dimension of $\mathfrak{h}_n(q)$ is exactly $3n$, while it becomes infinite when $q$ is a root of unity. We then demonstrate the rigidity of its iterated Ore extension structure, showing that any such presentation is essentially unique up to permutation and scaling of variables. The graded automorphism group is completely determined to be isomorphic to $(\mathbb{C}^*)^{2n} \rtimes S_n$. Furthermore, $\mathfrak{h}_n(q)$ is shown to possess a universal deformation property as the canonical PBW-preserving deformation of the classical Heisenberg algebra $\mathfrak{h}_n(1)$. We compute its Hilbert series as $(1-t)^{-3n}$, confirming polynomial growth of degree $3n$, and establish that its Gelfand--Kirillov dimension coincides with its classical Krull dimension. These results are extended to a generalized multi-parameter version $\mathfrak{H}_n(\mathbf{Q})$, and illustrated through detailed examples and applications in representation theory and deformation quantization.