Homological properties of some quantum Heisenberg algebras
Samuel A. Lopes, Héctor Suárez, Yésica Suárez
TL;DR
This work analyzes the homological properties of Heisenberg-type algebras, including classical, quantum, generalized, and quantum generalized variants, through the lens of skew PBW extensions and graded iterated Ore extensions. It establishes precise criteria for when generalized Heisenberg algebras are skew PBW extensions and when they are noetherian, showing that many favorable homological features (Koszulity, AS-regularity, Calabi-Yau-ness) occur only in specific degenerate cases (notably f = t). For quantum generalized Heisenberg algebras with f = pt and quadratic g, the authors prove Koszulity, AS-regularity, and graded Calabi–Yau property of dimension 3, and they identify these as Jacobian algebras of homogeneous potentials, with Calabi–Yau status preserved under PBW deformations. Overall, the paper provides a unified framework connecting structural extensions with robust homological properties across a broad family of Heisenberg-type algebras.
Abstract
In this paper we study the properties Koszul, Artin-Schelter regular and (skew) Calabi-Yau of some special types of quantum and generalized Heisenberg algebras and also analyze relations between these algebras, (graded) iterated Ore extensions and (graded) skew PBW extensions. The first-named author and Razavinia introduced the quantum generalized Heisenberg algebras, which depend on a parameter $q$ and two polynomials $f, g\in K[t]$. We prove that under certain conditions for $f$, $g$ these algebras are Koszul, Artin-Shelter regular, Calabi-Yau and graded Calabi-Yau.