# Leavitt path algebras: the first decade

@article{Abrams2014LeavittPA, title={Leavitt path algebras: the first decade}, author={G. Abrams}, journal={Bulletin of Mathematical Sciences}, year={2014}, volume={5}, pages={59-120} }

The algebraic structures known as Leavitt path algebras were initially developed in 2004 by Ara, Moreno and Pardo, and almost simultaneously (using a different approach) by the author and Aranda Pino. During the intervening decade, these algebras have attracted significant interest and attention, not only from ring theorists, but from analysts working in C∗-algebras, group theorists, and symbolic dynamicists as well. The goal of this article is threefold: to introduce the notion of Leavitt path… Expand

#### 59 Citations

The Groupoid Approach to Leavitt Path Algebras

- Mathematics
- 2018

When the theory of Leavitt path algebras was already quite advanced, it was discovered that some of the more difficult questions were susceptible to a new approach using topological groupoids. The… Expand

The Basics of Leavitt Path Algebras: Motivations, Definitions and Examples

- Mathematics
- 2017

We introduce the central idea, that of a Leavitt path algebra. We start by describing the classical Leavitt algebras. We then proceed to give the definition of the Leavitt path algebra L K (E) for an… Expand

Leavitt path algebras with coefficients in a Clifford semifield

- Mathematics
- 2019

Abstract In this article, we define the Leavitt path algebra of a directed graph Γ with coefficients in a Clifford semifield S. The general properties of are briefly discussed. Then, concentrating on… Expand

Simpleness of Leavitt path algebras with coefficients in a commutative semiring

- Mathematics
- 2015

In this paper, we study ideal- and congruence-simpleness for the Leavitt path algebras of directed graphs with coefficients in a commutative semiring S, establishing some fundamental properties of… Expand

Centers of Leavitt path algebras and their completions

- Mathematics
- 2015

In [8, 9] M. G. Corrales Garcia, D. M. Barquero, C. Martin Gonzalez, M. Siles Molina, J. F Solanilla Hernandez described the center of a Leavitt path algebra and characterized it in terms of the… Expand

Generalizations, Applications, and Current Lines of Research

- Mathematics
- 2017

We conclude the book with various observations regarding three important aspects of Leavitt path algebras. First, we describe various generalizations of, and constructions related to, Leavitt path… Expand

A Survey on the Ideal Structure of Leavitt Path Algebras

- Mathematics
- 2020

There is extensive recent literature on the graded, non-graded, prime, primitive, maximal ideals of Leavitt path algebras. In this introductory level survey, we will be giving an overview of… Expand

A Survey of Some of the Recent Developments in Leavitt Path Algebras

- Mathematics
- 2020

In this survey article, we describe some of recent ring-theoretic and module-theoretic investigations of a Leavitt path algebra L of an arbitrary directed graph E over a field K. It is shown how a… Expand

Lie nilpotent Novikov algebras and Lie solvable Leavitt path algebras

- Mathematics
- 2020

In this paper, we first study properties of the lower central chains for Novikov algebras. Then we show that for every Lie nilpotent Novikov algebra N , the ideal of N generated by the set {ab − ba |… Expand

Representations of Leavitt path algebras

- Mathematics
- 2015

We study representations of a Leavitt path algebra $L$ of a finitely separated digraph $\Gamma$ over a field. We show that the category of $L$-modules is equivalent to a full subcategory of quiver… Expand

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Let E be any directed graph, and K be any field. For any ideal I of the Leavitt path algebra LK(E) we provide an explicit description of a set of generators for I. This description allows us to… Expand

The classification question for Leavitt path algebras

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Leavitt path algebras with coefficients in a commutative ring

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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… Expand