Leavitt Path Algebras Overview

• Leavitt path algebras are noncommutative associative algebras defined from directed graphs and Cuntz–Krieger relations, generalizing classical Leavitt algebras.
• They exhibit the Bézout property where every finitely generated ideal is principal, with proofs extending from finite graphs to direct limits of arbitrary graphs.
• Their well-structured ideal theory and graded module characteristics play a crucial role in noncommutative ring theory, operator algebras, and algebraic K-theory.

A Leavitt path algebra is a noncommutative associative algebra constructed from a directed graph, encoding both the graph’s combinatorial structure and a set of Cuntz–Krieger relations. Originally introduced to generalize the classical Leavitt algebras of module type $(1,n)$, these algebras have become pivotal in noncommutative ring theory, symbolic dynamics, operator algebras, and algebraic $K$-theory. Defined over an arbitrary field $K$ and arbitrary graph $E$, $L_K(E)$ exhibits deep ideal-theoretic, module-theoretic, and regularity phenomena, with the canonical $\mathbb{Z}$-grading playing a crucial role.

1. Algebraic Construction and Universal Properties

Let $E=(E^0,E^1,s,r)$ be a directed graph, possibly infinite, with $E^0$ the set of vertices and $E^1$ the set of edges. The Leavitt path algebra $L_K(E)$ is generated by:

• Pairwise orthogonal idempotents $\{v: v \in E^0\}$
• Edges $\{e: e \in E^1\}$
• Ghost edges $\{e^*: e \in E^1\}$

Subject to the relations: $$\begin{aligned} \text{(V)}\quad& v w = \delta_{v,w}v &&\forall\,v,w\in E^0 \ \text{(E1)}\quad& s(e) e = e = e r(e) &&\forall\,e\in E^1 \ \text{(E2)}\quad& r(e) e^* = e^* = e^* s(e) &&\forall\,e\in E^1 \ \text{(CK1)}\quad& e^* f = \delta_{e,f} r(e) &&\forall\,e,f \in E^1 \ \text{(CK2)}\quad& v = \sum_{e\in s^{-1}(v)} e e^* && \forall\,v:\,0<|s^{-1}(v)|<\infty \end{aligned}$$ The algebra is equipped with a canonical $\mathbb{Z}$-grading via $\deg(v)=0$, $\deg(e)=1$, $\deg(e^*)=-1$, allowing all elements to be expressed as sums of monomials $p q^*$ where $p$ and $q$ are paths in $E$.

2. Bézout Property and Principal Ideals

The fundamental result of Abrams–Mantese–Tonolo is that every Leavitt path algebra $L_K(E)$ over any field and any directed graph is a Bézout ring (Abrams et al., 2016). That is, every finitely generated left or right ideal is principal: $$\forall\, I \subseteq L_K(E),\, I=\sum_{j=1}^m L_K(E)\, x_j \implies \exists\, x \in L_K(E):\; I = L_K(E) x$$ This result holds for both finite and infinite graphs and does not require any restriction on the field $K$.

Outline of proof:

• For finite $E$, induction on $|E^0|$ divides the proof into three cases:
  • No sources/cycles: UGN fails, so every finitely generated ideal is cyclic.
  • Source vertex: Reduce to smaller graphs by source elimination and apply induction.
  • Source cycle: Decompose the algebra into direct sums and corners of matrix algebras over $K[x,x^{-1}]$, which are principal ideal rings.
• For arbitrary $E$, $L_K(E)$ is a direct limit of Bézout algebras associated to finite subgraphs; the Bézout property passes to directed limits.

3. Ideal Theory and Multiplicative Structure

Given the Bézout property, every finitely generated two-sided ideal is principal. Furthermore, Leavitt path algebras are arithmetical rings: the lattice of two-sided ideals is distributive, i.e., for any ideals $A,B,C$,

$A \cap (B + C) = (A\cap B) + (A\cap C)$

They are also multiplication rings, meaning for $A \subseteq B$, there is always $C$ with $A=BC$ (Rangaswamy, 2016).

Commutativity of ideal multiplication holds: $AB=BA$ for all ideals. Ideals factor uniquely into products of prime ideals, and for finite graphs or when $L_K(E)$ is Artinian/Noetherian, every ideal decomposes as a finite product of primes. The irreducible and primary ideals coincide and are precisely the powers of primes.

4. Module-Theoretic Consequences and Projectives

In a Bézout ring, every cyclic projective module is principal. Every finitely generated projective module that embeds in the ring is generated by a single element. The monoid of isomorphism classes of finitely generated projective $L_K(E)$-modules is presented as: $$M_E = \left\langle\,v \ : \ v = \sum_{e \in s^{-1}(v)} r(e)\ \ (v\not\text{ sink}) \right\rangle$$ Finitely generated projective modules over $L_K(E)$ correspond bijectively to certain combinatorial data on the underlying graph.

5. Examples and Illustrations

Finite graphs:

• For $E$ with one vertex and one edge, $L_K(E) \cong K[x,x^{-1}]$ is a principal ideal domain.
• For $E$ with one vertex and $n$ loops, $L_K(E) \cong L_K(1,n)$, every finitely generated left ideal is cyclic.

Simple illustration: For the graph $E^0 = \{v, w\}$, $E^1 = \{f: v \to w\}$,

$L_K(E) \cong M_2(K)$

which is a classical principal ideal ring.

Ideals: In $L_K(E)$, the left ideal generated by $E_{22}$ in $M_2(K)$ is principal, as is any finitely generated one-sided ideal.

6. Structural and Field-Independence Remarks

The Bézout property is independent of the characteristic or cardinality of $K$. The proof leverages combinatorial reductions using sources and cycles in $E$ and generalizes smoothly to arbitrary graphs via direct limits.

This property streamlines structural investigations, e.g., injectivity, divisibility, or Baer properties, as divisibility and annihilator conditions only need to be checked for single generators.

7. Connections and Impact

The Bézout property for Leavitt path algebras complements deeper results on their regularity, flatness, and cancellation properties (Hazrat, 2013), as well as classifications via monoids, Morita theory, and $K$-theoretic invariants. It plays a fundamental role in the module-theoretic landscape of graph algebras and interacts richly with multiplicative ideal theory, refinement monoids, and ring-theoretic regularity.

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References:

Abrams, Mantese, Tonolo, "Leavitt path algebras are Bézout" (Abrams et al., 2016) Rangaswamy, "Multiplicative ideal theory of Leavitt path algebras" (Rangaswamy, 2016)