Homological properties of quantum generalized Heisenberg algebras

This paper presents a survey of several classes of Heisenberg-type algebras, including classical, quantum, and generalized variants. We examine the relationships between different families, such as quantum Heisenberg algebras, generalized Heisenberg algebras, quantum generalized Heisenberg algebras, Ore extensions and skew PBW extensions. We also explore homological properties of a special class of quantum generalized Heisenberg algebras. Special attention is given to the Koszul, Artin–Schelter regular, and (skew) Calabi–Yau properties of quantum generalized Heisenberg algebras.