Tame Leavitt Path Algebra

• Tame Leavitt path algebra is an algebraic structure derived from separated graphs that unifies concepts from Leavitt path algebras, inverse semigroups, and noncommutative rings.
• The unique Leavitt–Munn normal form provides an explicit K-basis by reducing elements to combinations of reduced Leavitt–Munn trees.
• It connects to partial crossed products, ample groupoids, and Steinberg algebras, offering broad applications in operator algebras and refinement monoids.

A tame Leavitt path algebra $\mathcal{L}_K^\text{ab}(E,C)$ is an algebraic object constructed from a separated graph $(E,C)$ that generalizes and unifies structures arising in Leavitt path algebras, inverse semigroup theory, and noncommutative ring theory. The construction utilizes the Leavitt inverse semigroup $\mathcal{LI}(E,C)$, which itself is defined as a quotient of the separated graph inverse semigroup $\mathcal{S}(E,C)$, imposing Cuntz--Krieger-type relations that correspond to specific path and separation structures in the graph. Tame Leavitt path algebras admit a normal form for their elements in terms of Leavitt--Munn trees and are closely related to partial semidirect products, ample groupoids, and Steinberg algebras. They play a central structural role in the realization and classification of various algebraic and analytic objects associated with (separated) graphs, including all finitely generated conical refinement monoids and a broad class of algebras and $C^*$-algebras (Ara et al., 17 Dec 2025, Ara et al., 8 Mar 2024, Ara et al., 2019).

1. Separated Graphs and Inverse Semigroup Foundations

A separated graph $(E,C)$ consists of a directed graph $E = (E^0, E^1, s, r)$ and, at each vertex $v \in E^0$, a partition $C_v$ of the incoming edges $s^{-1}(v)$ into pairwise-disjoint, nonempty subsets called separations. The full separation data is $C = \bigsqcup_{v \in E^0} C_v$. Notable special cases are the trivial separation (the classical graph case: $C_v = \{s^{-1}(v)\}$) and the free separation ($C_v = \{\{e\}: e \in s^{-1}(v)\}$).

The separated-graph inverse semigroup $\mathcal{S}(E,C)$ is the universal inverse semigroup with zero generated by idempotents corresponding to vertices and partial isometries (and their inverses) corresponding to edges, subject to relations encoding the graph structure, involution, and separation (notably, $x^{-1}y = \delta_{x,y}r(x)$ for $x,y$ in the same class $X \in C$) (Ara et al., 17 Dec 2025, Ara et al., 8 Mar 2024).

$\mathcal{S}(E,C)$ is always strongly $E^*$-unitary: there exists an idempotent-pure partial homomorphism into the free group $\mathbb{F}(E^1)$, yielding a partial crossed-product structure on $\mathcal{S}(E,C)$ (Ara et al., 8 Mar 2024).

2. Definition and Structure of the Leavitt Inverse Semigroup

The Leavitt inverse semigroup $\mathcal{LI}(E,C)$ is constructed as a quotient of $\mathcal{S}(E,C)$ by imposing the Cuntz–Krieger relations for singleton classes: $ee^{-1} = s(e)$ whenever $\{e\} \in C$. Thus,

$\mathcal{LI}(E,C) = \mathcal{S}(E,C)/\sim_L$

where the congruence is generated by these idempotent identities (Ara et al., 17 Dec 2025).

$\mathcal{LI}(E,C)$ admits a semidirect-product decomposition: it is strongly $E^*$-unitary with an idempotent-pure partial homomorphism

$$\omega\colon \mathcal{LI}(E,C)^{\times}\rightarrow \mathbb{F}(E^1)$$

so that

$$\mathcal{LI}(E,C) \cong \mathcal{E}(\mathcal{LI}(E,C)) \rtimes^r_{\theta} \mathbb{F}(E^1)$$

where $\mathcal{E}(\mathcal{LI}(E,C))$ is the semilattice of idempotents and $\theta$ is a system of partial semilattice automorphisms determined by the graph structure and separation (Ara et al., 17 Dec 2025, Ara et al., 8 Mar 2024).

Idempotents of $\mathcal{LI}(E,C)$ are parametrized by finite $C$-compatible subsets of the free groupoid of the graph (Leavitt--Munn trees), appropriately modded out by an equivalence relating certain singleton representatives.

3. Tame Leavitt Path Algebra: Presentation, Normal Form, and Basis

Given a commutative unital ring $K$, the tame Leavitt path algebra $\mathcal{L}_K^{\text{ab}}(E,C)$ is constructed as the $K$-algebra generated by the canonical images of generators of $\mathcal{LI}(E,C)$, subject to the same relations.

A key structural result is the existence of a unique normal form for elements of $\mathcal{L}_K^{\text{ab}}(E,C)$, known as the Leavitt--Munn normal form: $$a = \sum_{i} \lambda_i\bigl(e(T_i)\rtimes g_i\bigr)$$ where each $(T_i, g_i)$ is a Leavitt--Munn tree in reduced form and $e(T)$ is the idempotent corresponding to the finite $C$-compatible subset $T$ of the free groupoid (Ara et al., 17 Dec 2025).

The set $$\{e(T) \rtimes g : T \in D^L_g,\, T \text{ reduced}\}$$ forms a $K$-basis for $\mathcal{L}_K^{\text{ab}}(E,C)$. The proof utilizes the Diamond Lemma and careful reduction/rewriting strategies in the core subalgebra generated by idempotents (Ara et al., 17 Dec 2025).

4. Connections with Partial Actions, Groupoid, and Steinberg Algebra Models

The structure of $\mathcal{LI}(E,C)$ and $\mathcal{L}_K^{\text{ab}}(E,C)$ enables their realization as partial crossed products and algebras of étale groupoids. There is a canonical topological groupoid (the tight groupoid $\mathcal{G}_{\text{tight}}(E,C)$) associated to the tight spectrum of the idempotent semilattice, such that

$$\mathcal{L}_K^{\text{ab}}(E,C) \cong A_K(\mathcal{G}_{\text{tight}}(E,C))$$

where $A_K$ denotes the Steinberg algebra of locally constant, compactly supported $K$-valued functions under convolution (Ara et al., 17 Dec 2025, Ara et al., 8 Mar 2024, Ara et al., 2019).

These constructions extend and unify groupoid and algebraic models for Leavitt path algebras and their generalizations to separated graphs.

5. Kernel, Socle, and Spectral Properties

There is a natural exact sequence

$$0 \longrightarrow \mathcal{Q} \longrightarrow \mathcal{C}_K^{\text{ab}}(E,C) \longrightarrow \mathcal{L}_K^{\text{ab}}(E,C) \longrightarrow 0$$

where $\mathcal{C}_K^{\text{ab}}(E,C)$ is the tame Cohn algebra and $\mathcal{Q}$ is generated by projections $q_X = v - \sum_{e \in X} ee^*$ for each finite class $X \in C$. A basis for the kernel $\mathcal{Q}$ is given by "blocked trees," i.e., elements of the form $e(T \setminus F) \rtimes g$ for finite blocking families $F$ (Ara et al., 17 Dec 2025).

The socle of $\mathcal{L}_K^{\text{ab}}(E,C)$ decomposes as

$$\operatorname{soc}(\mathcal{L}_K^{\text{ab}}(E,C)) = \bigoplus_{[T] \in \Omega_{\text{iso}}/\mathcal R} M_{|[T]|}(K)$$

where $\Omega_{\text{iso}}$ is the set of Leavitt trees without exits and with trivial isotropy, and $\mathcal R$ is orbit equivalence (Ara et al., 17 Dec 2025).

6. Key Examples and Applications

The framework recovers several important algebras and semigroup constructions:

• Non-separated graphs: The classical Leavitt path algebras, socle generated by line-point vertices, normal forms of Abrams–Aranda Pino–Siles Molina.
• Cuntz separated graphs: For a one-vertex, free separation, $\mathcal{LI}(E,C) \cong \mathbb{F}(E^1)$, and $$\mathcal{L}_K^{\text{ab}}(E,C) \cong K[\mathbb{F}(E^1)]$$.
• Free inverse monoid algebras: Realized as corners in tame Cohn algebras associated to separated graphs with free separation structure; basis described via blocked Munn trees.

The construction of these algebras and their socles, idempotent semilattices, and actions provides a means to realize all finitely generated conical refinement monoids and solves realization problems in von Neumann regular rings (Ara et al., 2019).

7. Structural Implications and Generalizations

The tame Leavitt path algebra serves as a unifying object for a wide spectrum of structures arising from directed graphs, inverse semigroups, and groupoid algebras. Through its semidirect product decomposition, groupoid model, and Steinberg algebra realization, it encompasses and extends classical Leavitt path algebras, their $C^*$-algebra analogs, and associated refinement monoids. The amenability, ideal structure, and $K$-theory of the resulting crossed product algebras are accessible via the underlying groupoid topology and the partial action structure (Ara et al., 17 Dec 2025, Ara et al., 8 Mar 2024, Ara et al., 2019).

The theory provides explicit combinatorial and algebraic tools—normal forms, bases, projection ideals, and socle decompositions—for comprehensive analysis, thereby deepening the interaction between semigroup theory, operator algebras, and graph-theoretic algebraic structures.