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

TL;DR

This paper thoroughly analyzes the structural properties of the $q$-Heisenberg algebra, including homology, automorphisms, and deformations, revealing key invariants and rigidity features depending on the parameter $q$.

Contribution

It provides a comprehensive structural analysis of the $q$-Heisenberg algebra, including homological dimensions, automorphism groups, and deformation properties, extending to multi-parameter versions.

Findings

Homological dimension is $3n$ when $q$ is not a root of unity.

Automorphism group is isomorphic to $(C^*)^{2n} times S_n$.

The algebra has a universal deformation property and polynomial growth with Hilbert series $(1-t)^{-3n}.

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.