Leavitt path algebras with coefficients in a commutative ring

By M. Tomforde. 2009

109 citations recorded by the source corpus.

Given a directed graph E we describe a method for constructing a Leavitt path algebra $L_R(E)$ whose coefficients are in a commutative unital ring R. We prove versions of the Graded Uniqueness Theorem and Cuntz-Krieger Uniqueness Theorem for these Leavitt path algebras, giving proofs that both generalize and simplify the classical results for Leavitt path algebras over fields. We also analyze the ideal structure of $L_R(E)$, and we prove that if $K$ is a field, then $L_K(E) \cong K \otimes_\Z L_\Z(E)$.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/58a533c85e95732c1d7aeba4f92691853f98449a

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Aidan Sims
2. L. O. Clark
3. Toke Meier Carlsen
4. Iain Raeburn
5. Efren Ruiz
6. G. Pino
7. Astrid an Huef
8. John Clark

In Geometry and Topology

1. Aidan Sims
2. Efren Ruiz
3. Toke Meier Carlsen
4. Iain Raeburn
5. G. Pino
6. Astrid an Huef
7. Cain Edie-Michell
8. L. O. Clark

In Mathematical Physics

1. Aidan Sims
2. Efren Ruiz
3. Toke Meier Carlsen
4. Iain Raeburn
5. L. O. Clark
6. Astrid an Huef
7. John Clark
8. G. Pino

Citations in the corpus

• Leavitt path algebras: the first decade (2014, unknown): “[98].”
• Kumjian-Pask algebras of higher-rank graphs (2011, methodology): “However, since we are allowing coefficients in a commutative ring, we have to follow [38] and restrict attention to the basic ideals, which are the ideals I such that rpv ∈ I and r ∈ R \ {0} imply pv ∈ I.”
• Equivalence and stable isomorphism of groupoids, and diagonal-preserving stable isomorphisms of graph C*-algebras and Leavitt path algebras (2016, methodology): “We assume familiarity with graph C-algebras and Leavitt path algebras; see [6] and [31] for the requisite background.”
• Uniqueness Theorems for Steinberg Algebras (2014, methodology): “Steinberg algebras also include the Kumjian-Pask algebras of [1] and [5] which themselves include the Leavitt path algebras of [9].”