arXiv subjects

Abdourrahmane Kabbaj

Publications and source records attributed to Abdourrahmane Kabbaj.

2 recordsLinked to original sources

arXiv2025

On GK Dimension and Generator Bounds for a Class of Graded Algebras

In this paper, we introduce the concept of \textit{monotonic algebras}, a broad class of algebras that includes all Artin-Schelter regular algebras of dimension at most four, as well as algebras with \textit{pure} resolutions, such as Koszul and piecewise Koszul algebras. We show that the Gelfand-Kirillov (GK) dimension of these algebras is bounded above by their global dimension and establish a similar result for the minimal number of generators. Furthermore, we prove a parity theorem for Artin-Schelter regular algebras, demonstrating that the difference between their global dimension and GK dimension is always an even integer.

math.RA↗

arXiv2024

Where are the N-Koszul algebras of finite global dimension?

The class of $N$-Koszul graded algebras of finite global dimension has gained lots of attention in recent years, especially in the study of Artin-Schelter regular algebras. While structurally rich and concrete, the only known examples of such algebras are either when $N = 2$, i.e. the algebra is Koszul, or when $N = 3$. Under a mild Hilbert series assumption, we rule out the existence of $N$-Koszul graded algebras of finite global dimension for $N$ not prime. Furthermore, we establish strong restrictions on the global dimension of such algebras. This suggests that perhaps the existence of 3-Koszul algebras with finite global dimension and `nice' Hilbert series is an anomaly.

math.RA↗