On differential smoothness of certain Artin-Schelter regular algebras of dimension 5

TL;DR

This paper studies the differential smoothness of five-dimensional Artin-Schelter regular algebras, identifying structural obstructions and providing examples of both smooth and non-smooth cases.

Contribution

It establishes criteria for differential smoothness in 5D AS-regular algebras and presents explicit examples illustrating these properties.

Findings

Two- and four-generator algebras lack differential calculus

A five-generator graded Clifford algebra is differentially smooth

Structural obstructions depend on generator number and algebra type

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.