The derived dimensions and representation distances of artin algebras

TL;DR

This paper introduces a new class of algebras called $(m,n)$-Igusa-Todorov algebras, providing methods for their construction and exploring their implications for derived dimensions and representation distances in artin algebras.

Contribution

It generalizes Igusa-Todorov algebras to $(m,n)$-Igusa-Todorov algebras and establishes their role in relating derived dimensions to representation distances.

Findings

Constructed methods for $(m,n)$-Igusa-Todorov algebras

Established relationship between derived dimension and representation distance

Provided improved upper bounds for derived dimension in certain classes

Abstract

There is a well-known class of algebras called Igusa-Todorov algebras which were introduced in relation to finitistic dimension conjecture. As a generalization of Igusa-Todorov algebras, the new notion of $(m,n)$-Igusa-Todorov algebras provides a wider framework for studying derived dimensions. In this paper, we give methods for constructing $(m,n)$-Igusa-Todorov algebras. As an application, we present for general artin algebras a relationship between the derived dimension and the representation distance. Moreover, we end this paper to show that the main result can be used to give a better upper bound for the derived dimension for some classes of algebras.