Leavitt Path Algebras with Coefficients in a Commutative Unital Ring

TL;DR

This paper extends known properties of Leavitt path algebras from fields to commutative rings, introduces new structural results, and provides formulas for algebraic dimensions based on the underlying digraphs.

Contribution

It generalizes Leavitt path algebra properties to commutative rings and introduces new results on ideal lattices, bases, and Gelfand-Kirillov dimension formulas.

Findings

Ideal lattice of Leavitt path algebra embeds into that of the path algebra

Constructs a new basis for polynomial growth Leavitt path algebras

Provides a formula for Gelfand-Kirillov dimension in terms of the digraph

Abstract

In addition to extending some facts from field coefficients to commutative ring coefficients for Leavitt path algebras with new shorter proofs, we also prove some results that are new even for field coefficients. In particular, we show that the ideal lattice of a Leavitt path algebra embeds into the ideal lattice of the path algebra of the same digraph, we construct a new basis for a Leavitt path algebra of polynomial growth and give a formula for the Gelfand-Kirillov dimension of a Leavitt path algebra in terms of its digraph.