On $n$-cubic Pyramid Algebras

TL;DR

This paper introduces a new class of algebras called $n$-cubic pyramid algebras, characterizes their projective resolutions, and connects them to existing structures like Iyama's $n$-Auslander algebras, revealing their periodicity and almost Koszul properties.

Contribution

It defines $n$-cubic pyramid algebras, characterizes their projective resolutions via $n$-cuboids, and links them to Iyama's cone construction, expanding understanding of $n$-representation theory.

Findings

Projective resolutions characterized by $n$-cuboids

Algebras are periodic and almost Koszul

Recovers Iyama's cone construction using $n$-cubic pyramid algebras

Abstract

In this paper we study a class of algebras having $n$-dimensional pyramid shaped quiver with $n$-cubic cells, which we called $n$-cubic pyramid algebras. This class of algebras includes the quadratic dual of the basic $n$-Auslander absolutely $n$-complete algebras introduced by Iyama. We show that the projective resolution of the simples of $n$-cubic pyramid algebras can be characterized by $n$-cuboids, and prove that they are periodic. So these algebras are almost Koszul and $(n-1)$-translation algebras. We also recover Iyama's cone construction for $n$-Auslander absolutely $n$-complete algebras using $n$-cubic pyramid algebras and the theory of $n$-translation algebras.