Grothendieck group of the Leavitt path algebra over power graphs of prime-power cyclic groups

TL;DR

This paper computes the Grothendieck group of Leavitt path algebras over power graphs of prime-power cyclic groups using Smith normal form, providing algebraic insights into these structures.

Contribution

It introduces a method to determine the Grothendieck group for these algebras via linear algebra techniques, specifically Smith normal form calculations.

Findings

Explicit computation of the Grothendieck group for the algebra

Application of Smith normal form to adjacency matrices

Enhanced understanding of algebraic invariants of power graph-based algebras

Abstract

In this paper, the Grothendieck group of the Leavitt path algebra over the power graphs of all prime-power cyclic groups is studied by using a well-known computation from linear algebra. More precisely, the Smith normal form of the matrix derived from the adjacency matrix associated with the power graph of prime-power cyclic group is calculated.