Coverings and Truncations of Graded Selfinjective Algebras

TL;DR

This paper explores the structure of graded self-injective algebras, their smash products, and related constructions like $ au$-slice algebras, revealing new relationships and methods for constructing algebras with finite global dimension.

Contribution

It introduces $ au$-slice algebras with finite global dimension, shows their trivial extensions are isomorphic, and connects these to Iyama's absolute $n$-complete algebra via truncation.

Findings

$ au$-slice algebras have finite global dimension.

Trivial extensions of $ au$-slice algebras are isomorphic.

Revealed relationships between smash products, Beilinson algebra, and Iyama's algebra.

Abstract

Let $\Lambda$ be a graded self-injective algebra. We describe its smash product $\Lambda# k\mathbb Z^*$ with the group $\mathbb Z$, its Beilinson algebra and their relationship. Starting with $\Lambda$, we construct algebras with finite global dimension, called $\tau$-slice algebras, we show that their trivial extensions are all isomorphic, and their repetitive algebras are the same $\Lambda# k\mathbb Z^*$. There exist $\tau$-mutations similar to the BGP reflections for the $\tau$-slice algebras. We also recover Iyama's absolute $n$-complete algebra as truncation of the Koszul dual of certain self-injective algebra.