This survey of the recent developments in the investigations of a Leavitt path algebra L of an arbitrary graph E over a field K consists of two parts. In the first part describes how very often a single graph property of E implies multiple ring properties of L, thus making Leavitt path algebras effective tools in constructing rings of various desirable properties. The second part describes methods of constructing simple modules over L and characterizes Leavitt path algebras all of whose simple modules possess some specific properties such as being flat, finitely presented,being graded, injective etc.
Let $K$ be a field. We characterise the row-finite weighted graphs $(E,w)$ such that the weighted Leavitt path algebra $L_K(E,w)$ is isomorphic to an unweighted Leavitt path algebra. Moreover, we prove that if $L_K(E,w)$ is locally finite, or Noetherian, or Artinian, or von Neumann regular, or has finite Gelfand-Kirillov dimension, then $L_K(E,w)$ is isomorphic to an unweighted Leavitt path algebra.
Weighted Leavitt path algebras were introduced in 2013 by Roozbeh Hazrat. These algebras generalise simultaneously the usual Leavitt path algebras and William Leavitt's algebras $L(m,n)$. In this paper we try to give an overview of what is known about the weighted Leavitt path algebras. We also prove some new results (in particular on the graded K-theory of weighted Leavitt path algebras) and mention open problems.
We achieve an extremely useful description (up to isomorphism) of the Leavitt path algebra $L_K(E)$ of a finite graph $E$ with coefficients in a field $K$ as a direct sum of matrix rings over $K$, direct sum with a corner of the Leavitt path algebra $L_K(F)$ of a graph $F$ for which every regular vertex is the base of a loop. Moreover, in this case one may transform the graph $E$ into the graph $F$ via some step-by-step procedure, using the "source elimination" and "collapsing" processes. We use this to establish the main result of the article, that every nonzero corner of a Leavitt path algeb…
In this article, we give a new class of automorphisms of Leavitt path algebras of arbitrary graphs. Consequently, we obtain Anick type automorphisms of these Leavitt path algebras and new irreducible representations of Leavitt algebras of type $(1, n)$.
We show that every graded ideal of a Leavitt path algebra is graded isomorphic to a Leavitt path algebra. It is known that a graded ideal $I$ of a Leavitt path algebra is isomorphic to the Leavitt path algebra of a graph, known as the generalized hedgehog graph, which is defined based on certain sets of vertices uniquely determined by $I$. However, this isomorphism may not be graded. We show that replacing the short "spines" of the generalized hedgehog graph with possibly fewer, but then necessarily longer spines, we obtain a graph (which we call the porcupine graph) such that its Leavitt path…
We define Leavitt path algebras of hypergraphs generalizing simultaneously Leavitt path algebras of finitely separated graphs and Leavitt path algebras of row-finite vertex-weighted graphs. We find linear bases for those algebras, compute their Gelfand-Kirillov dimension, obtain some results on ring-theoretic properties like simplicity, von Neumann regularity and Noetherianess and investigate their K-theory and graded K-theory. By doing so we obtain new results on the Gelfand-Kirillov dimension and graded K-theory of Leavitt path algebras of separated graphs and on the graded K-theory of weigh…
Leavitt path algebras associate to directed graphs a $\mathbb Z$-graded algebra and in their simplest form recover the Leavitt algebras L(1,n). In this note, we introduce iterated Leavitt path algebras associated to directed weighted graphs which have natural $\oplus \mathbb Z$ grading and in their simplest form recover the Leavitt algebras $L(n,k)$. We also characterize Leavitt path algebras which are strongly graded.
We describe the centroid of some Leavitt path algebras. More precisely, we show that for Leavitt path algebras over a field $K$ that are simple its centroid is isomorphic to $K$, and for prime Leavitt path algebras its centroid is isomorphic to $K$ except if the graph is a row-finite comet, in which case the centroid is isomorphic to $K[x,x^{-1}]$.
Leavitt path algebras associate to directed graphs a $\mathbb Z$-graded algebra and in their simplest form recover the Leavitt algebras $L(1,k)$. In this note, we first study this $\mathbb Z$-grading and characterize the ($\mathbb Z$-graded) structure of Leavitt path algebras, associated to finite acyclic graphs, $C_n$-comet and multi-headed graphs. The last two type are examples of graphs whose Leavitt path algebras are strongly graded. We characterize Leavitt path algebras which are strongly graded, along the way obtaining classes of algebras which are group rings or crossed-products. In an …
We determine the Gelfand-Kirillov dimension of a weighted Leavitt path algebra $L_K(E,w)$ where $K$ is a field and $(E,w)$ a finite weighted graph. Further we show that a finite-dimensional weighted Leavitt path algebra over a field $K$ is isomorphic to a finite product of matrix rings over $K$.
We compute the monoid of isomorphism classes of finitely generated projective modules of a Leavitt path algebra over an arbitrary directed graph. Our result generalizes the result of Ara, Moreno, and Pardo in which they computed this monoid of a Leavitt path algebra over a countable row-finite directed graph.
We introduce a class of topologies on the Leavitt path algebra $L(Γ)$ of a finite directed graph and decompose a graded completion $\widehat{L}(Γ)$ as a direct sum of minimal ideals.
When twist of a Leavitt path algebra is again a Leavitt path algebra
Institute of Mathematics, Vietnam Academy of Science and Technology（越南科学技术院数学研究所）；Department of Mathematics and Statistics, Saint Louis University（圣路易斯大学数学与统计系）
We show that, for an arbitrary graph, a regular ideal of the associated Leavitt path algebra is also graded. As a consequence, for a row-finite graph, we obtain that the quotient of the associated Leavitt path by a regular ideal is again a Leavitt path algebra and that Condition~(L) is preserved by quotients by regular ideals. Furthermore, we describe the vertex set of a regular ideal and make a comparison between the theory of regular ideals in Leavitt path algebras and in graph C*-algebras.
In sharp contrast to the Abrams-Rangaswamy Theorem that the only von Neumann regular Leavitt path algebras are exactly those associated to acyclic graphs, here we prove that the Leavitt path algebra of any arbitrary graph is a graded von Neumann regular ring. Several properties of Leavitt path algebras, such as triviality of the Jacobson radical, flatness of graded modules and finitely generated graded right (left) ideals being generated by an idempotent element, follow as a consequence of general theory of grade von Neumann regular rings.
We compute the $V$-monoid of a weighted Leavitt path algebra of a row-finite weighted graph, correcting a wrong computation of the $V$-monoid that exists in the literature. Further we show that the description of $K_0$ of a weighted Leavitt path algebra that exists in the literature is correct (although the computation was based on a wrong $V$-monoid description).
In this paper we characterize the minimal left ideals of a Leavitt path algebra as those ones which are isomorphic to principal left ideals generated by line point vertices, that is, by vertices whose trees do not contain neither bifurcations nor closed paths. Moreover, we show that the socle of a Leavitt path algebra is the two-sided ideal generated by these line point vertices. This characterization allows us to compute the socle of some algebras that arise as the Leavitt path algebra of some row-finite graphs. A complete description of the socle of a Leavitt path algebra is given: it is a l…
There is an 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 different types of ideals and the correspondence between the lattice of ideals and the lattice of hereditary and saturated subsets of the graph over which the Leavitt path algebra is constructed.
The primary purpose of this thesis is to show every ultragraph Leavitt path algebra is Morita equivalent, as a ring, to a graph Leavitt path algebra. Takeshi Katsura, Paul Muhly, Aidan Sims, and Mark Tomforde showed every ultragraph $C^{*}$-algebra is Morita equivalent, in the $C^{*}$-sense, to a graph $C^{*}$-algebra; our result is an algebraic analog of this fact. Further, we will use our result to give an alternate proof for established conditions which guarantee the simplicity of an ultragraph Leavitt path algebra over a field.
In this paper a bijection between the set of prime ideals of a Leavitt path algebra $L_K(E)$ and a certain set which involves maximal tails in $E$ and the prime spectrum of $K[x,x^{-1}]$ is established. Necessary and sufficient conditions on the graph $E$ so that the Leavitt path algebra $L_K(E)$ is primitive are also found.
In this paper we show that Leavitt path algebras of weighted graphs and Leavitt path algebras of separated graphs are intimately related. We prove that any Leavitt path algebra $L(E,ω)$ of a row-finite vertex weighted graph $(E,ω)$ is $*$-isomorphic to the lower Leavitt path algebra of a certain bipartite separated graph $(E(ω),C(ω))$. For a general locally finite weighted graph $(E, ω)$, we show that a certain quotient $L_1(E,ω)$ of $L(E,ω)$ is $*$-isomorphic to an upper Leavitt path algebra of another bipartite separated graph $(E(w)_1,C(w)^1)$. We furthermore introduce the algebra $L^{\math…
For a field $F$ and a row-finite directed graph $Γ$ let $L(Γ)$ be the Leavitt path algebra. We find necessary and sufficient conditions for the Lie algebra $[L(Γ),L(Γ)]$ to be simple.
In this paper, we give a matrix-theoretic criterion for the Leavitt path algebra of a finite graph has Invariant Basis Number. Consequently, we show that the Cohn path algebra of a finite graph has Invariant Basis Number, as well as provide some certain classes of finite graphs for which Leavitt path algebras having Invariant Basis Number.
In this paper, we prove that the multiplicative group of a unital non-commutative Leavitt path algebra $L_K(E)$ and Cohn path algebra $C_K(E)$ contain a non-cyclic free subgroup, provided $K$ is a non-absolute field. We also provide a description of the generators of free subgroups in term of the graph $E$. Finally, we determine multiplicative groups of Leavitt path algebras of some special types.
We give necessary and sufficient conditions on a row-finite graph E so that the Leavitt path algebra L(E) is purely infinite simple. This result provides the algebraic analog to the corresponding result for the Cuntz-Krieger C$^*$-algebra C$^*$(E) given in [7].
For any field K and directed graph E, we completely describe the elements of the Leavitt path algebra L_K(E) which lie in the commutator subspace [L_K(E),L_K(E)]. We then use this result to classify all Leavitt path algebras L_K(E) that satisfy L_K(E)=[L_K(E),L_K(E)]. We also show that these Leavitt path algebras have the additional (unusual) property that all their Lie ideals are (ring-theoretic) ideals, and construct examples of such rings with various ideal structures.
We provide a characterization of graded von Neumann regular rings involving the recently introduced class of nearly epsilon-strongly graded rings. As our main application, we generalize Hazrat's result that Leavitt path algebras over fields are graded von Neumann regular. More precisely, we show that a Leavitt path algebra $L_R(E)$ with coefficients in a unital ring $R$ is graded von Neumann regular if and only if $R$ is von Neumann regular. We also prove that both Leavitt path algebras and corner skew Laurent polynomial rings over von Neumann regular rings are semiprimitive and semiprime. The…
This is a short note on how a particular graph construction on a subset of edges that lead to a subalgebra construction, provided a tool in proving some ring theoretical properties of Leavitt path algebras.
For any positive integer $n$ we describe the Leavitt path algebra of the Cayley graph $C_n$ corresponding to the cyclic group $\Z/n\Z$. Using a Kirchberg-Phillips-type realization result, we show that there are exactly four isomorphism classes of such Leavitt path algebras, arising as the algebras corresponding to the graphs $C_i$ ($3\leq i \leq 6$).
A ring R is called right principally-injective if every R-homomorphism from a principal right ideal aR to R (a in R), extends to R, or equivalently if every system of equations xa=b (a, b in R) is solvable in R. In this paper we show that for any arbitrary graph E and for a field K, principally-injective conditions for the Leavitt path algebra LK(E) are equivalent to that the graph E being acyclic. We also show that the principally injective Leavitt path algebras are precisely the von Neumann regular Leavitt path algebras.
The purpose of this paper is to provide a common framework for studying various generalizations of Leavitt algebras and Leavitt path algebras. This paper consists of two parts. In part I we define Cohn-Leavitt path algebras of a new class of graphs with an additional structure called bi-separated graphs, which generalize the constructions of Leavitt path algebras of various types of graphs. We define and study the category \textbf{BSG} of bi-separated graphs with appropriate morphisms so that the functor which associates a bi-separated graph to its Cohn-Leavitt path algebra is continuous. We a…
In this article, we give necessary and sufficient conditions under which the Leavitt path algebra $L_K(\mathcal{G})$ of an ultragraph $\mathcal{G}$ over a field $K$ is purely infinite simple and that it is von Neumann regular. Consequently, we obtain that every graded simple ultragraph Leavitt path algebra is either a locally matricial algebra, or a full matrix ring over $K[x, x^{-1}]$, or a purely infinite simple algebra.
For any unital commutative ring $R$ and for any graph $E$, we identify the commutative core of the Leavitt path algebra of $E$ with coefficients in $R$, which is a maximal commutative subalgebra of the Leavitt path algebra. Furthermore, we are able to characterize injectivity of representations which gives a generalization of the Cuntz-Krieger uniqueness theorem.
The stable rank of Leavitt path algebras of row-finite graphs was computed by Ara and Pardo. In this paper we extend this for an arbitrary directed graph. In some parts, we proceed our computation as the row-finite case while in some parts we use the knowledge about row-finite setting by applying the desingularizing method due to Drinen and Tomforde. In particular, we characterize purely infinite simple quotients of a Leavitt path algebra.
In this paper we completely describe graphically Leavitt path algebras with bounded index of nilpotence and show that each graded simple module $S$ over a Leavitt path algebra with bounded index of nilpotence is graded $Σ$-injective, that is, $S^{(α)}$ is graded injective for any cardinal $α$. Furthermore, we characterize Leavitt path algebras over which each simple module is $Σ$-injective. We have shown that each simple module over a Leavitt path algebra $L_K(E)$ is $Σ$-injective if and only if the graph $E$ contains no cycles, and there is a positive integer $d$ such that the length of any p…
We explicitly describe two-sided ideals in Leavitt path algebras associated with a row-finite graph. Our main result is that any two-sided ideal $I$ of a Leavitt path algebra associated with a row-finite graph is generated by elements of the form $v + \sum_{i=1}^nλ_i g^i$, where $g$ is a cycle based at vertex $v$. We use this result to show that a Leavitt path algebra is two-sided Noetherian if and only if the ascending chain condition holds for hereditary and saturated closures of the subsets of the vertices of the row-finite graph $E$.
We present a result of P. Ara which establishes that the Unbounded Generating Number property is a Morita invariant for unital rings. Using this, we give necessary and sufficient conditions on a graph $E$ so that the Leavitt path algebra associated to $E$ has UGN. We conclude by identifying the graphs for which the Leavitt path algebra is (equivalently) directly finite; stably finite; Hermite; and has cancellation of projectives.
We characterize Leavitt path algebras which are exchange rings in terms of intrinsic properties of the graph and show that the values of the stable rank for these algebras are 1, 2 or $\infty$. Concrete criteria in terms of properties of the underlying graph are given for each case.
Let $E$ be a directed graph, $K$ any field, and let $L_K(E)$ denote the Leavitt path algebra of $E$ with coefficients in $K$. We show that $L_K(E)$ is a Bézout ring, i.e., that every finitely generated one-sided ideal of $L_K(E)$ is principal.
A group graded $K$-algebra $A=\bigoplus\limits_{g\in G} A_g$ is called "locally finite" if $\dim_K A_g < \infty$ for every $g\in G$. We characterise the weighted graphs $(E,w)$ for which the weighted Leavitt path algebra $L_K(E,w)$ is locally finite with respect to its standard grading. We also prove that the locally finite weighted Leavitt path algebras are precisely the Noetherian ones and that $L_K(E,w)$ is locally finite iff $(E,w)$ is finite and the Gelfand-Kirillov dimension of $L_K(E,w)$ equals $0$ or $1$. Further it is shown that a locally finite weighted Leavitt path algebra is isomor…
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 underlying graph. We offer a different characterization of the center. In particular, we prove that the Boolean algebra of central idempotents \ of a Leavitt path algebra of a finite graph is isomorphic to the Boolean algebra of finitary annihilator hereditary subsets of the graph.
Ideals in Leavitt path algebras have been shown to share many properties with those of integral domains. Since studying factorizations of ideals in integral domains into special types of ideals (particularly, prime, prime-power, primary, irreducible, semiprime, and quasi-primary ones) has proved fruitful, we conduct an analogous investigation in the context of Leavitt path algebras. Specifically, we classify the proper ideals in these rings that admit factorizations into products of each of the above types of ideals. We also classify the Leavitt path algebras where every proper ideal admits a …
We study the center of several types of path algebras. We start with the path algebra $KE$ and prove that if the number of vertices is infinite then the center is zero. Otherwise, it coincides with the field $K$ except when the graph $E$ is a cycle in which case the center is $K[x]$, the polynomial algebra in one indeterminate. Then we compute the centers of prime Cohn and Leavitt path algebras. A lower and an upper bound for the center of a Leavitt path algebra are given by introducing the graded Baer radical for graded algebras.
In this paper, we study ideal- and congruence-simpleness for the Leavitt path algebras of directed graphs with coefficients in a commutative semiring S, as well as establish some fundamental properties of those algebras. We provide a complete characterization of ideal-simple Leavitt path algebras with coefficients in a semifield S that extends the well-known characterizations when the ground semiring S is a field. Also, extending the well-known characterizations when S is a field or commutative ring, we present a complete characterization of congruence-simple Leavitt path algebras over row-fin…
For any field $K$ and for a completely arbitrary graph $E$, we characterize the Leavitt path algebras $L_K(E)$ that are indecomposable (as a direct sum of two-sided ideals) in terms of the underlying graph. When the algebra decomposes, it actually does so as a direct sum of Leavitt path algebras for some suitable graphs. Under certain finiteness conditions, a unique indecomposable decomposition exists.
In this paper, we give sharp bounds for the homological dimensions of the Leavitt path algebra $L_K(E)$ of a finite graph $E$ with coefficients in a commutative ring $K$, as well as establish a formula for calculating the homological dimensions of $L_K(E)$ when $K$ is a commutative unital algebra over a field.
We investigate conditions under which the endomorphism ring of the Leavitt path algebra $L_{K}(E)$ possesses various ring and module-theoretical properties such as being von Neumann regular, $π$-regular, strongly $π$-regular or self-injective. We also describe conditions under which $L_{K}(E)$ is continuous as well as automorphism invariant as a right $L_{K}(E)$-module.
