The Igusa-Todorov $\phi$ function for truncated path algebras

TL;DR

This paper investigates the Igusa-Todorov $$ function for truncated path algebras, establishing symmetry of $$-dimension, computing it in relation to simpler algebras, and characterizing algebras with $$-dimension one.

Contribution

It proves the symmetry of $$-dimension for truncated path algebras and provides formulas and characterizations related to $$-dimension.

Findings

$idim A = idim A^{ ext{op}}$ for truncated path algebras

Bounds $$-dimension based on $$-dimension of simpler algebras

Characterizes algebras with $$-dimension equal to 1

Abstract

Given a truncated path algebra $A=\frac{\Bbbk Q}{J^k}$ we prove that $\fidim A = \fidim A^{\op}$. We also compute the $\phi$-dimension of $A$ in function of the $\phi$-dimension of $\frac{\Bbbk Q}{J^2}$ when $Q$ has no sources nor sinks. This allows us to bound the $\phi$-dimension for truncated path algebras. Finally, we characterize $A$ when its $\phi$-dimension is equal to $1$.