具有不相交循环的图的 Leavitt 路代数上的内射模

Leavitt path algebras: Graded direct-finiteness and graded $Σ$-injective simple modules

In this paper, we give a complete characterization of Leavitt path algebras which are graded $Σ$-$V$ rings, that is, rings over which a direct sum of arbitrary copies of any graded simple module is graded injective. Specifically, we show that a Leavitt path algebra $L$ over an arbitrary graph $E$ is a graded $Σ$-$V$ ring if and only if it is a subdirect product of matrix rings of arbitrary size but with finitely many non-zero entries over $K$ or $K[x,x^{-1}]$ with appropriate matrix gradings. We also obtain a graphical characterization of such a graded $Σ$-$V$ ring $L$% . When the graph $E$ is…

莱维特路代数上单模的内射包络I：单左理想

The injective envelope of simple modules over Leavitt path algebras I: simple left ideals

本文刻画莱维特路代数单左理想，显式构造其内射包络，将雅可比代数相关构造推广至所有图的所有单左理想，还给出两类无限发射点对应单模内射包络的描述。

Prüfer modules over Leavitt path algebras

Let $L_K(E)$ denote the Leavitt path algebra associated to the finite graph $E$ and field $K$. For any closed path $c$ in $E$, we define and investigate the uniserial, artinian, non-noetherian left $L_K(E)$-module $U_{E,c-1}$. The unique simple factor of each proper submodule of $U_{E,c-1}$ is isomorphic to the Chen simple module $V_{[c^\infty]}$. In our main result, we classify those closed paths $c$ for which $U_{E,c-1}$ is injective. In this situation, $U_{E,c-1}$ is the injective hull of $V_{[c^\infty]}$.

Leavitt路代数上单模的内射包络 II：由排他圈产生的单模

The injective envelope of simple modules over Leavitt path algebras II: simples arising from exclusive cycles

本文针对Leavitt路代数上由排他圈产生的单模，构造无穷级数K-向量空间上的模结构，给出其内射包络的显式刻画，将相关结果推广至所有有限或无限图及所有排他圈。

Endomorphism rings of Leavitt path algebras

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.

A survey of some of the recent developments in Leavitt path algebras

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, injecti…

Injective modules over the Jacobson algebra $K< X,Y | XY = 1 >$

For a field $K$, let $\mathcal{R}$ denote the Jacobson algebra $K\langle X, Y \ | \ XY=1\rangle$. We give an explicit construction of the injective envelope of each of the (infinitely many) simple left $\mathcal{R}$-modules. Consequently, we obtain an explicit description of a minimal injective cogenerator for $\mathcal{R}$. Our approach involves realizing $\mathcal{R}$ up to isomorphism as the Leavitt path $K$-algebra of an appropriate graph $\mathcal{T}$, which thereby allows us to utilize important machinery developed for that class of algebras.

Graphs with disjoint cycles, classification via the talented monoid

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…

On graded irreducible representations of Leavitt path algebras

Using the E-algebraic systems, various graded irreducible representations of a Leavitt path algebra L of a graph E over a field K are constructed. The concept of a Laurent vertex is introduced and it is shown that the minimal graded left ideals of L are generated by Laurent vertices and/or line points leading to a detailed description of the graded socle of L. Following this, a complete characterization is obtained of the Leavitt path algebras over which every graded irreducible representation is finitely presented. A useful result is that the irreducible representation V_[p] induced by infini…

Canonical traces and directly finite Leavitt path algebras

Motivated by the study of traces on graph $C^*$-algebras, we consider traces (additive, central maps) on Leavitt path algebras, the algebraic counterparts of graph $C^*$-algebras. In particular, we consider traces which vanish on nonzero graded components of a Leavitt path algebra and refer to them as {\em canonical} since they are uniquely determined by their values on the vertices. A desirable property of a $\mathbb{C}$-valued trace on a $C^*$-algebra is that the trace of an element of the positive cone is nonnegative. We adapt this property to traces on a Leavitt path algebra $L_K(E)$ with …

The Dixmier-Moeglin equivalence for Leavitt path algebras

Let $K$ be a field, let $E$ be a finite directed graph, and let $L_K(E)$ be the Leavitt path algebra of $E$ over $K$. We show that for a prime ideal $P$ in $L_K(E)$, the following are equivalent: \begin{enumerate} \item $P$ is primitive; \item $P$ is rational; \item $P$ is locally closed in ${\rm Spec}(L_K(E))$. \end{enumerate} We show that the prime spectrum ${\rm Spec}(L_K(E))$ decomposes into a finite disjoint union of subsets, each of which is homeomorphic to ${\rm Spec}(K)$ or to ${\rm Spec}(K[x,x^{-1}])$. In the case that $K$ is infinite, we show that $L_K(E)$ has a rational $K^{\times}$…

Leavitt路代数的Baer环与Baer *-环刻画

Baer and Baer *-ring characterizations of Leavitt path algebras

该研究通过底图性质刻画了Leavitt路代数的Rickart、Baer及Baer $*$-环等零化子相关性质，推广至局部幺元环与分次情形，提供了环实例构造方法并对比了其与图$C^*$-代数的性质差异。

Homotopy categories, Leavitt path algebras and Gorenstein projective modules

For a finite quiver without sources or sinks, we prove that the homotopy category of acyclic complexes of injective modules over the corresponding finite dimensional algebra with radical square zero is triangle equivalent to the derived category of the Leavitt path algebra viewed as a differential graded algebra with trivial differential, which is further triangle equivalent to the stable category of Gorenstein projective modules over the trivial extension algebra of a von Neumann regular algebra by an invertible bimodule. A related, but different, result for the homotopy category of acyclic c…

Irreducible representations of Leavitt algebras

For a weighted graph $E$, we construct representation graphs $F$, and consequently, $L_K(E)$-modules $V_F$, where $L_K(E)$ is the Leavitt path algebra associated to $E$, with coefficients in a field $K$. We characterise representation graphs $F$ such that $V_F$ are simple $L_K(E)$-modules. We show that the category of representation graphs of $E$, $RG(E)$, is a disjoint union of subcategories, each of which contains a unique universal object $T$ which gives an indecomposable $L_K(E)$-module $V_T$ and a unique irreducible representation graph $S$, which gives a simple $L_K(E)$-module $V_S$. Spe…

分离图的Leavitt逆半群

The Leavitt inverse semigroup of a separated graph

本文研究了分离图的Leavitt逆半群，提供了有限Leavitt路径代数的标准逆半群模型，并通过规范形式和结构性质证明了其单射性。

Williams猜想对流星图成立

Williams' Conjecture holds for meteor graphs

本文证明流星图的移位等价、强移位等价、Leavitt路径代数的分次Morita等价及分次K-理论模同构彼此等价，并由此建立Leavitt路径代数与图C*-代数在Morita等价意义下的对应关系。

带不交环图的动力学及其应用

Dynamics on graphs with disjoint cycles and applications

本文研究带不交环图的动力学，通过分裂变换得到标准形式，证明环长两两互素的3环流星图的四个等价条件成立，进而验证两个猜想对这类图成立。

有向图中不相交环的格分类猜想成立

The Graded Classification Conjecture holds for graphs with disjoint cycles

本文证明了有限图中不相交环的格分类猜想成立，并通过特定图操作建立了图代数的格$*$-同构类。