A new characterisation of quasi-hereditary Nakayama algebras and applications

TL;DR

This paper characterizes quasi-hereditary Nakayama algebras through a new property called S-connectedness, linking algebraic structure with projective dimensions and Fibonacci number enumeration.

Contribution

It establishes that S-connectedness is equivalent to being quasi-hereditary for Nakayama algebras and classifies cases where an inequality for global dimension is tight.

Findings

S-connectedness characterizes quasi-hereditary Nakayama algebras.

The paper improves an existing inequality for global dimension.

Enumerates specific Nakayama algebras using Fibonacci numbers.

Abstract

We call a finite dimensional algebra A S-connected if the projective dimensions of the simple A-modules form an interval. We prove that a Nakayama algebra A is S-connected if and only if A is quasi-hereditary. We apply this result to improve an inequality for the global dimension of quasi-hereditary Nakayama algebras due to Brown. We furthermore classify the Nakayama algebras where equality is attained in Brown's inequality and show that they are enumerated by the even indexed Fibonacci numbers if the algebra is cyclic and by the odd indexed Fibonacci numbers if the algebra is linear.