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

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.

💡 Research Summary

This paper provides a comprehensive study of the $q$‑Heisenberg algebra $\mathfrak h_n(q)$, a quantum deformation of the classical Heisenberg algebra, focusing on five interrelated structural aspects: global homological dimension, rigidity of its iterated Ore‑extension presentation, graded automorphism group, universal deformation property, and Hilbert series/Gelfand–Kirillov growth.

1. Global homological dimension.
Using the known PBW basis and the iterated Ore‑extension description, the authors first establish that when $q$ is not a root of unity, $\mathfrak h_n(q)$ is an Auslander‑regular, Cohen–Macaulay, and Artin–Schelter regular algebra of dimension $3n$. Consequently the global (left and right) homological dimension equals exactly $3n$. In contrast, if $q$ is a root of unity, central elements $z_i^{\ell}$ (with $q^\ell=1$) become non‑regular, forcing the global dimension to be infinite. This dichotomy sharpens earlier partial results and aligns with known behavior of quantum algebras at roots of unity.

2. Rigidity of the Ore‑extension structure.
The algebra can be written as
\