On differential smoothness of certain Artin-Schelter regular algebras of dimension 5
Andrés Rubiano
TL;DR
The paper investigates differential smoothness for five-dimensional Artin–Schelter regular algebras, linking the GK-dimension to generator count. It proves a general obstruction (NoDS) that, under a generator-wise reflective automorphism framework, fails for algebras with $\mathrm{GKdim}=m$ and $\mathrm{Gen}=n$ when $m>n$, thereby blocking $m$-dimensional connected calculi in many cases. This obstruction applies to known 5D AS-regular algebras generated by two or four elements, highlighting a fundamental mismatch between generators and GK-dimension in differential smoothness. In contrast, the paper constructs a differentiably smooth example: a five-generator graded Clifford algebra $C$, with an explicit $5$-dimensional differential calculus, volume form, and integrable structure, confirming differential smoothness in this high-generator case. Together, these results clarify how generator count vs. GK-dimension governs the existence of noncommutative calculi and inform the landscape of five-dimensional AS-regular algebras in noncommutative geometry.
Abstract
This article investigates the differential smoothness of various five-dimensional Artin-Schelter regular algebras. By analyzing the relationship between the number of generators and the Gelfand-Kirillov dimension, we provide structural obstructions to differential smoothness in specific algebraic families. In particular, we prove that certain two- and four-generator AS-regular algebras of global dimension five fail to admit a differential calculus, while a five-generator graded Clifford algebra provides a contrasting positive example.