Singular equivalence of finite dimensional algebras with radical square zero

TL;DR

This paper demonstrates that certain quiver operations preserve the singular equivalence of associated finite dimensional algebras with radical square zero, unifying various natural operations in symbolic dynamics under this framework.

Contribution

It proves that Crisp and Gow's quiver operation yields singularly equivalent algebras, generalizing several natural operations in symbolic dynamics.

Findings

Quiver operations can preserve singular equivalence of algebras.

The operation includes strong shift equivalence, splitting, and source elimination.

Singular equivalence is maintained across a broad class of quiver modifications.

Abstract

We prove that the Crisp and Gow's quiver operation on a finite quiver Q produces a new quiver Q' with fewer vertices, such that the finite dimensional algebras kQ/J^2 and kQ'/J^2 are singularly equivalent. This operation is a general quiver operation which includes as specific examples some operations which arise naturally in symbolic dynamics (e.g., (elementary) strong shift equivalent, (in-out) splitting, source elimination, etc.).