Some New Weighted Leavitt Path Algebra Homomorphisms

An Honours Thesis Summary

Motivation and Background

In the 1950s, W. G. Leavitt made a surprising discovery: there exist rings that violate a property we take for granted for vector spaces. For a field K, if K^m ≅ Kⁿ as K-vector spaces, then m = n. More generally, a ring R has the Invariant Basis Number (IBN) property if R^m ≅ Rⁿ as (left) R-modules implies m = n. Leavitt constructed rings R for which R ≅ Rⁿ for some n ≥ 2 (in particular R ≅ R ⊕ R), showing that IBN can fail.

In 2004, two independent groups, Abrams and Aranda Pino on one side, and Ara, Moreno, and Pardo on the other, discovered that Leavitt's constructions could be vastly generalised by associating algebras to directed graphs. The result was Leavitt path algebras (LPAs), a rich theory connecting graph combinatorics, ring theory, and even operator algebras (via their analytic cousins, graph C*-algebras).

The central insight is this: given a directed graph and a field K, one can define a K-algebra using the graph's vertices and edges as generators, subject to certain relations. Different graphs yield algebras with dramatically different properties: some are simple matrix rings, others are exotic non-IBN rings, and the graph structure determines which.

Why Graphs Are the Right Objects

Leavitt path algebras are defined via generators and relations. Given a directed graph E:

Generators are the vertices, edges, and "ghost edges" (formal inverses of edges).
Relations encode how these pieces interact.

The relations ensure that: 1. Vertices act as orthogonal idempotents (they multiply to zero unless identical). 2. Edges "connect" vertices appropriately. 3. Ghost edges reverse the direction. 4. At regular (non-sink) vertices, the outgoing edges partition the identity.

What makes graphs so effective is that they provide a visual calculus for algebra. Properties of the algebra, simplicity, the IBN property, graded structure, can often be read directly from the graph. For instance: - Acyclic graphs yield finite-dimensional matrix algebras. - Petal graphs (a single vertex with loops) yield Leavitt's original non-IBN examples. - The presence of cycles with exits leads to richer, more subtle behaviour.

The Graphs Studied in This Thesis

The thesis focuses on several families of graphs, all relatively small but illuminating:

Oriented line graphs: A chain of vertices connected by edges pointing in one direction, ending at a sink. The LPA of an n-vertex line graph is isomorphic to the ring of n × n matrices over the base field.

Petal graphs: A single vertex with multiple loop edges. The n-petal graph yields an algebra R where R ≅ Rⁿ, the quintessential non-IBN example.

Weighted graphs: A generalisation where edges carry positive integer weights. A structured edge of weight k behaves like k parallel edges but with coupled relations. This captures Leavitt algebras of arbitrary type (m, n).

The thesis pays particular attention to:
- A weighted graph with one vertex and two loops of weights 1 and 2. This was an open problem case at the time of writing, where it was unknown whether the algebra is isomorphic to any unweighted LPA.
- A weighted graph with two vertices and bidirectional edges of weights 1 and 2, introduced in the thesis to construct new homomorphisms.
- An unweighted graph with three vertices arranged in a specific pattern, which admits an embedding into the open-problem weighted algebra.

Main Ideas and Results

The Core Question

Weighted Leavitt path algebras (wLPAs) generalise ordinary LPAs. A natural question: when is a wLPA isomorphic to an ordinary LPA?

For many weighted graphs, one can construct an explicit unweighted graph whose LPA matches. But for one particular example, a single-vertex graph with loops of weight 1 and 2, this question was open. My thesis investigates the structure of this algebra by constructing homomorphisms to and from related algebras.

Computational Approach

A significant part of my work involved developing software in Haskell to manipulate LPA elements. The key features:

Equality checking: LPAs are defined by relations, so determining when two expressions represent the same element is non-trivial. I implemented an algorithm that rewrites expressions by eliminating "forbidden" subpaths until reaching a canonical form.
Homomorphism verification: To show a map is a well-defined homomorphism, one must verify that all relations in the domain map to zero in the codomain. My software automates these checks.
Basis enumeration: The "nod-path" basis (paths containing no forbidden subwords) provides a canonical spanning set. The software can enumerate these.

The New Homomorphisms

I constructed several new homomorphisms connecting weighted and unweighted LPAs:

An injection from the open-problem wLPA into a larger wLPA: Starting from the single-vertex weighted graph, I embed its algebra into a two-vertex weighted graph with bidirectional edges. The map sends the single vertex to the sum of two vertices and distributes the loops across the edge structure.

A surjection onto a known wLPA: From my two-vertex weighted graph, there is an epimorphism onto a simpler two-vertex weighted graph whose algebra is known to be isomorphic to an unweighted LPA. I determined the kernel explicitly: it is generated by three specific elements relating the "backward" edges to adjoints of "forward" edges.

The composition: Combining these gives a homomorphism from the open-problem algebra into the known algebra. The kernel of this composition is also explicitly determined.

An embedding of an unweighted LPA: Perhaps most surprisingly, a certain three-vertex unweighted graph (with one vertex having a loop) embeds into the open-problem weighted algebra. The embedding is given by explicit formulas involving products and adjoints of the weighted generators.

Matrix Representations

I found that certain families of weighted graphs admit representations in matrix rings over the open-problem algebra:

A chain of n vertices with bidirectional weighted edges embeds into n × n matrices over the single-vertex weighted algebra.
By composing with a monomorphism into a Laurent polynomial ring extension, one obtains representations in matrices over unweighted LPAs with Laurent polynomial coefficients.

These representations provide concrete ways to compute with these algebras and suggest structural relationships between different weighted graphs.

How This Work Fits Into the Literature

This thesis builds on work by several researchers:

Hazrat introduced weighted LPAs and developed much of their foundational theory.
Preusser studied the V-monoid of weighted LPAs and posed the open question about the single-vertex graph with weights 1 and 2.
Classical work on Leavitt algebras, graph monoids, and graded rings provides the theoretical backdrop.

My contributions are: 1. New explicit homomorphisms that illuminate the structure of the open-problem algebra. 2. Computational verification of these homomorphisms. 3. A demonstration that the open-problem algebra contains interesting subalgebras (specifically, an unweighted LPA) and embeds into algebras with additional structure (Laurent polynomial extensions).

While I did not resolve whether the open-problem wLPA is isomorphic to an unweighted LPA, the homomorphisms I constructed provide new angles of attack and show that the algebra has rich internal structure.

What I Learned and What Comes Next

Lessons Learned

The power of computation: Abstract algebra can feel impenetrable. Quotients of free algebras by ideals are not objects you can easily manipulate by hand. Writing software to automate calculations was transformative. It let me experiment, test conjectures, and verify claims that would be tedious or error-prone to check manually.

The subtlety of well-definedness: When working with algebras defined by generators and relations, the most basic question, "is this map well-defined?", requires checking that relations map to zero. This is conceptually simple but computationally intensive.

The interplay between graphs and algebra: The ability to translate between graph-theoretic properties (acyclicity, exits, weights) and algebraic properties (IBN, graded structure, ideal structure) is the heart of LPA theory. Developing intuition for this correspondence took time but was deeply rewarding.

Open Problems and Future Directions

Several claims in my thesis are stated without complete proofs:

Injectivity: I claim certain maps are injective but did not prove this rigorously. One approach, suggested by Preusser, is to show that nod-paths map to distinct nod-paths.
Termination: My algorithm for computing canonical forms has not been proven to terminate. In practice it always does, but a formal proof would establish that nod-paths genuinely span the algebra.
The open problem: Does the wLPA of the single-vertex graph with weights 1 and 2 admit an isomorphism to some unweighted LPA? My work provides new constraints but not a resolution.
Generalisation: Can my matrix representation results be extended to broader families of weighted graphs? There seem to be patterns worth exploring.