It is a conjecture that for the class of Leavitt path algebras associated to finite directed graphs, their graded Grothendieck groups $K_0^{\mathrm{gr}}$ are a complete invariant. For a Leavitt path algebra $L_{\mathsf k}(E)$, with coefficient in a field ${\mathsf k}$, the monoid of the positive cone of $K_0^{\mathrm{gr}}(L_{\mathsf k}(E))$ can be described completely in terms of the graph $E$. In this note we further investigate the structure of this "talented monoid", showing how it captures intrinsic properties of the graph and hence the structure of its associated Leavitt path algebras. In particular, for the class of strongly connected graphs, we show that the notion of the period of a graph can be completely described via the talented monoid. As an application, we will give a finer characterisation of the purely infinite simple Leavitt path algebras in terms of properties of the associated graph. We show that graded isomorphism of algebras preserve the period of the graphs, and obtain results giving more evidence to the graded classification conjecture.
We characterise directed graphs consisting of disjoint cycles via their talented monoids. We show that a graph $E$ consists of disjoint cycles precisely when its talented monoid $T_E$ has a certain Jordan-Hölder composition series. These are graphs whose associated Leavitt path algebras have finite Gelfand-Kirillov dimension (GKdim). We show that this dimension can be determined as the length of certain ideal series of the talented monoid. Since $T_E$ is the positive cone of the graded Grothendieck group $K_0^{gr}(L_K (E))$, we conclude that for graphs $E$ and $F$, if $K_0^{gr}(L_K (E))\cong K…
In this article we establish relationships between Leavitt path algebras, talented monoids and the adjacency matrices of the underlying graphs. We show that indeed the adjacency matrix generates in some sense the group action on the generators of the talented monoid. With the help of this we deduce a form of the aperiodicity index of a graph via the talented monoid. We classify hereditary and saturated subsets via the adjacency matrix. Moreover we give a formula to compute all paths of a given length in a Leavitt path algebra based on the adjacency matrix. In addition we discuss the number of …
In this article, we study properties as simplicity, solvability and nilpotency for Lie bracket algebras arising from Leavitt path algebras, based on the talented monoid of the underlying graph. We show that graded simplicity and simplicity of the Leavitt path algebra can be connected via the Lie bracket algebra. Moreover, we use the Gelfand-Kirillov dimension for the Leavitt path algebra for a classification of nilpotency and solvability.
There is a tight relation between the geometry of a directed graph and the algebraic structure of a Leavitt path algebra associated to it. In this note, we show a similar connection between the geometry of the graph and the structure of a certain monoid associated to it. This monoid is isomorphic to the positive cone of the graded K0-group of the Leavitt path algebra which is naturally equipped with a Z-action. As an example, we show that a graph has a cycle without an exit if and only if the monoid has a periodic element. Consequently a graph has Condition (L) if and only if the group Z acts …
Some more talents of the talented monoid of a higher-rank graph
Western Sydney University（西悉尼大学）；Center for Pure Mathematics, School of Mathematical Sciences, Anhui University（安徽大学数学科学学院纯数学中心）；Department of Mathematics, Jadavpur University（贾达普大学数学系）
In this paper we consider endomorphisms of a finite directed path from monoid generators perspective. Our main aim is to determine the rank of the monoid $\wEnd\vec{P}_n$ of all weak endomorphisms of a directed path with $n$ vertices, which is a submonoid of the widely studied monoid $\On$ of all order-preserving transformations of a $n$-chain. Also, we describe the regular elements of $\wEnd\vec{P}_n$ and calculate its size and number of idempotents.
Let $Γ$ be the infinite cyclic group on a generator $x.$ To avoid confusion when working with $\mathbb Z$-modules which also have an additional $\mathbb Z$-action, we consider the $\mathbb Z$-action to be a $Γ$-action instead. Starting from a directed graph $E$, one can define a cancellative commutative monoid $M_E^Γ$ with a $Γ$-action which agrees with the monoid structure and a natural order. The order and the action enable one to label each nonzero element as being exactly one of the following: comparable (periodic or aperiodic) or incomparable. We comprehensively pair up these element feat…
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.
In our main result, we establish that any conical sandpile monoid $M = SP(G)$ of a directed sandpile graph $G$ can be realised as the $\mathcal{V}$-monoid of a weighted Leavitt path algebra $L_K(E,w)$, and consequently, the sandpile group as the Grothendieck group $K_0(L_K(E,w))$. We show how to explicitly construct $(E,w)$ from $G$. Additionally, we describe the conical sandpile monoids which arise as the $\mathcal{V}$-monoid of a standard (i.e., unweighted) Leavitt path algebra.
Motivated by reconstruction results by Rubin, we introduce a new reconstruction notion for permutation groups, transformation monoids and clones, called automatic action compatibility, which entails automatic homeomorphicity. We further give a characterization of automatic homeomorphicity for transformation monoids on arbitrary carriers with a dense group of invertibles having automatic homeomorphicity. We then show how to lift automatic action compatibility from groups to monoids and from monoids to clones under fairly weak assumptions. We finally employ these theorems to get automatic action…
Let $E$ be a finite directed graph, and let $I$ be the poset obtained as the antisymmetrization of its set of vertices with respect to a pre-order $\le$ that satisfies $v\le w$ whenever there exists a directed path from $w$ to $v$. Assuming that $I$ is a tree, we define a poset of fields over $I$ as a family $\mathbf K = \{ K_i :i\in I \}$ of fields $K_i$ such that $K_i\subseteq K_j$ if $j\le i$. We define the concepts of a Leavitt path algebra $L_{\mathbf K} (E)$ and a regular algebra $Q_{\mathbf K}(E)$ over the poset of fields $\mathbf K$, and we show that $Q_{\mathbf K}(E)$ is a hereditary …
An introduction to Leavitt path algebras of arbitrary directed graphs is presented, and direct limit techniques are developed, with which many results that had previously been proved for countable graphs can be extended to uncountable ones. Such results include characterizations of simplicity, characterizations of the exchange property, and cancellation conditions for the K-theoretic monoid of equivalence classes of idempotent matrices.
We obtain a complete structural characterization of Cohn-Leavitt algebras over no-exit objects as graded involutive algebras. Corollaries of this result include graph-theoretic conditions characterizing when a Leavitt path algebra is a directed union of (graded) matricial algebras over the underlying field and over the algebra of Laurent polynomials and when the monoid of isomorphism classes of finitely generated projective modules is atomic and cancellative. We introduce the non-unital generalizations of graded analogues of noetherian and artinian rings, graded locally noetherian and graded l…
The construction of the Leavitt path algebra associated to a directed graph $E$ is extended to incorporate a family $C$ consisting of partitions of the sets of edges emanating from the vertices of $E$. The new algebras, $L_K(E,C)$, are analyzed in terms of their homology, ideal theory, and K-theory. These algebras are proved to be hereditary, and it is shown that any conical abelian monoid occurs as the monoid $\mon{L_K(E,C)}$ of isomorphism classes of finitely generated projective modules over one of these algebras. The lattice of trace ideals of $L_K(E,C)$ is determined by graph-theoretic da…
To every directed graph $E$ one can associate a \emph{graph inverse semigroup} $G(E)$, where elements roughly correspond to possible paths in $E$. These semigroups generalize polycylic monoids, and they arise in the study of Leavitt path algebras, Cohn path algebras, Cuntz-Krieger $C^*$-algebras, and Toeplitz $C^*$-algebras. We investigate topologies that turn $G(E)$ into a topological semigroup. For instance, we show that in any such topology that is Hausdorff, $G(E)\setminus \{0\}$ must be discrete for any directed graph $E$. On the other hand, $G(E)$ need not be discrete in a Hausdorff semi…
