G. Aranda-Pino, E. Pardo, M. Siles-Molina
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.
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We characterize the values of the stable rank for Leavitt path algebras, by giving concrete criteria in terms of properties of the underlying graph.
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.
Cancellation properties of graded and nonunital rings. Graded clean and graded exchange Leavitt path algebras
本文将单位环的消去相关性质推广到非单位环和分次环，研究分次clean与分次exchange在分次矩阵环下的行为，并刻画Leavitt路径代数为分次clean或分次exchange的图论条件。
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 identify largest ideals in Leavitt path algebras: the largest locally left/right artinian (which is the largest semisimple one), the largest locally left/right noetherian without minimal idempotents, the largest exchange, and the largest purely infinite. This last ideal is described as a direct sum of purely infinite simple pieces plus purely infinite non-simple and non-decomposable pieces. The invariance under ring isomorphisms of these ideals is also studied.
On Leavitt path algebras of Hopf graphs
本文研究Hopf图的Leavitt路径代数，利用群G和分歧数据𝔯刻画其Gelfand-Kirillov维数、稳定秩、纯无限单性及非零有限维表示的存在性。
In this paper we completely describe graphically Leavitt path algebras with bounded index of nilpotence. We show that the Leavitt path algebra $L_{K}(E)$ has index of nilpotence at most $n$ if and only if no cycle in the graph $E$ has an exit and there is a fixed positive integer $n$ such that the number of distinct paths that end at any given vertex $v$ (including $v$, but not including the entire cycle $c$ in case $v$ lies on $c$) is less than or equal to $n$. Interestingly, the Leavitt path algebras having bounded index of nilpotence turn out to be precisely those that satisfy a polynomial …
Refining an idea of Rosenmann and Rosset we show that the now widely studied classical Leavitt algebra $L_K(1,n)$ over a field $K$ is a ring of right quotients of the unital free associative algebra of rank $n$ with respect to the perfect Gabriel topology defined by powers of an ideal of codimension 1, providing a conceptual, variable-free description of $L_K(1, n)$. This result puts Leavitt (path) algebras on the frontier of important research areas in localization theory, free ideal rings and their automorphism groups, quiver algebras and graph operator algebras. As applications one obtains …
We show that $E$ is a finite graph with no sinks if and only if the Leavitt path algebra $L_R(E)$ is isomorphic to an algebraic Cuntz-Krieger algebra if and only if the $C^*$-algebra $C^*(E)$ is unital and $rank(K_0(C^*(E)))=rank(K_1(C^*(E)))$. When $k$ is a field and $rank(k^{\times})< \infty$, we show that the Leavitt path algebra $L_k(E)$ is isomorphic to an algebraic Cuntz-Krieger algebra if and only if $L_k(E)$ is unital and $rank(K_1(L_k(E)))=(rank(k^{\times})+1)rank(K_0(L_k(E)))$. We also show that any unital $k$-algebra which is Morita equivalent or stably isomorphic to an algebraic Cu…
We introduce higher-rank analogues of the Leavitt path algebras, which we call the Kumjian-Pask algebras. We prove graded and Cuntz-Krieger uniqueness theorems for these algebras, and analyze their ideal structure.
The Kumjian-Pask algebras are path algebras associated to higher-rank graphs, and generalize the Leavitt path algebras. We study the center of simple Kumjian-Pask algebras and characterize commutative Kumjian-Pask algebras.
The Kumjian-Pask algebra of a higher-rank graph generalises the Leavitt path algebra of a directed graph. We extend the definition of Kumjian-Pask algebra to row-finite higher-rank graphs $Λ$ with sources which satisfy a local-convexity condition. After proving versions of the graded-uniqueness theorem and the Cuntz-Krieger uniqueness theorem, we study the Kumjian-Pask algebra of rank-2 Bratteli diagrams by studying certain finite subgraphs which are locally convex. We show that the desourcification procedure of Farthing and Webster yields a row-finite higher-rank graph $\tildeΛ$ without sourc…
On the identity component of a strongly graded ring - with applications to graph algebras
本文研究了Leavitt代数的零分量结构，证明其为自由积代数的直接极限，并展示了其具有IBN性质。
Rank conditions and amenability for rings associated to graphs
研究有向图对应的路径环、Cohn路径环和Leavitt路径环，给出当系数环R满足秩条件或强秩条件时，图需满足的充要条件，并应用于刻画可均路径代数的图。
Chain Conditions for Etale Groupoid Algebras
本文针对艾达尔群胚代数，刻画了其范畴诺特/阿廷、局部诺特/阿廷及半单性质，推广了Leavitt路代数的相关链条件研究结果。
The Kumjian-Pask algebra KP(Λ) is a graded algebra associated to a higher-rank graph Λand is a generalization of the Leavitt path algebra of a directed graph. We analyze the minimal left-ideals of KP(Λ), and identify its socle as a graded ideal by describing its generators in terms of a subset of vertices of the graph. We characterize when KP(Λ) is semisimple, and obtain a complete structure theorem for a semisimple Kumjian-Pask algebra.
We introduce a generalisation of Condition (K) to finitely separated graphs and show that it is equivalent to essential freeness of the associated partial action as well as the exchange property of any of the associated tame algebras. As a consequence, we can show that any tame separated graph algebra with the exchange property is separative.
Given a unital ring $R$ and a two-sided ideal $I$ of $R$, we consider the question of determining when a unit of $R/I$ can be lifted to a unit of $R$. For the wide class of separative exchange ideals $I$, we show that the only obstruction to lifting invertibles relies on a K-theoretic condition on $I$. This allows to extend previously known index theories to this context. Using this we can draw consequences for von Neumann regular rings and C*-algebras with real rank zero.
Skew-symmetric non-integer matrices with real entries can be viewed as quivers with non-integer weights of arrows. One can mutate such quivers according to usual rules of quiver mutation. Felikson and Tumarkin show that rank 3 mutation-finite non-integer quivers admit geometric realisations by partial reflections. This allows to define a notion of seeds (as Y-seeds), and hence, to define the exchange graphs for mutation classes. In this paper we study exchange graphs of mutation-finite quivers in rank 3. The concept of finite and affine type generalises naturally to non-integer quivers. In par…
For any (unital) exchange ring $R$ whose finitely generated projective modules satisfy the separative cancellation property ($A\oplus A\cong A\oplus B\cong B\oplus B$ implies $A\cong B$), it is shown that all invertible square matrices over $R$ can be diagonalized by elementary row and column operations. Consequently, the natural homomorphism $GL_1(R) \to K_1(R)$ is surjective. In combination with a result of Huaxin Lin, it follows that for any separative, unital C*-algebra $A$ with real rank zero, the topological $K_1(A)$ is naturally isomorphic to the unitary group $U(A)$ modulo the connecte…
In this note we extend the notion of greedy bases developed by Lee, Li, and Zelevinsky to rank two generalized cluster algebras, i.e. binomial exchange relations are replaced by polynomial exchange relations. In the process we give a combinatorial construction in terms of a refined notion of compatible pairs on a maximal Dyck path.
Sign-coherence and tropical sign pattern for rank $3$ real cluster-cyclic exchange matrices
本文研究了秩3实簇循环交换矩阵的c向量符号一致性，证明了其与二次方程根的关系，并展示了对应的交换图结构及二面体群的簇实现。
The assignment (nonstable K_0-theory), that to a ring R associates the monoid V(R) of Murray-von Neumann equivalence classes of idempotent infinite matrices with only finitely nonzero entries over R, extends naturally to a functor. We prove the following lifting properties of that functor: (1) There is no functor F, from simplicial monoids with order-unit with normalized positive homomorphisms to exchange rings, such that VF is equivalent to the identity. (2) There is no functor F, from simplicial monoids with order-unit with normalized positive embeddings to C*-algebras of real rank 0 (resp.,…
Exchange graphs and Ext-quivers of hearts of tube categories
本文通过引入A型预单简集对管范畴有界导出范畴中的心进行分类，证明其交换图连通，并分类诸心的Ext箭图，进而推出Bridgeland稳定性条件空间连通。
In this paper we find the exchange graph of the rank n binomial Laurent phenomenon algebra associated to the complete graph on n vertices. More specifically, we prove that this exchange graph is isomorphic to that of the rank n linear Laurent phenomenon algebra associated to the complete graph on n vertices discussed in arxiv.org/abs/1206.2612.
Replacing invertibility with quasi-invertibility in Bass' first stable range condition we discover a new class of rings, the QB-rings. These constitute a considerable enlargement of the class of rings with stable rank one (B-rings), and include examples like the ring of endomorphisms of a vector space over a field F, and the ring of all row- and column- finite matrices over F. We show that the category of QB-rings is stable under the formation of corners, ideals and quotients, as well as matrices and direct limits. We also give necessary and sufficient conditions for an extension of QB-rings t…
Pseudo grading on cluster automorphism group with application to cluster algebras of rank $3$
本文提出簇自同构群上的伪 $\mathbb{N}$-分次，证明其由 $G_0\cup G_1$ 生成，并据此完全确定了秩 $3$ 且交换矩阵不可分解的簇代数的簇自同构群。
Higher-rank graphs and the graded $K$-theory of Kumjian-Pask algebras
本文建立了与高阶图（Kumjian-Pask代数）相关的Leavitt代数的分次K-理论基础，通过证明无限路群胚的分次零阶同调与Kumjian-Pask代数的分次Grothendieck群之间的同构，并展示图操作保持分次K-理论，为分类提供了潜在工具。
It is proved that the exchange property, the Bass stable rank and the quasi-Bass property are all preserved under surjective inverse limits. This is then applied to multiplier rings by showing that in many cases can be obtained as inverse limits.
We study properties of minimal mutation-infinite quivers. In particular we show that every minimal-mutation infinite quiver of at least rank 4 is Louise and has a maximal green sequence. It then follows that the cluster algebras generated by these quivers are locally acyclic and hence equal to their upper cluster algebra. We also study which quivers in a mutation-class have a maximal green sequence. For any rank 3 quiver there are at most 6 quivers in its mutation class that admit a maximal green sequence. We also show that for every rank 4 minimal mutation-infinite quiver there is a finite co…
Let $Σ$ be Laurent phenomenon (LP) seed of rank $n$, $\mathcal{A}(Σ)$, $\mathcal{U}(Σ)$ and $\mathcal{L}(Σ)$ be its corresponding Laurent phenomenon algebra, upper bound and lower bound respectively. We prove that each seed of $\mathcal{A}(Σ)$ is uniquely defined by its cluster, and any two seeds of $\mathcal{A}(Σ)$ with $n-1$ common cluster variables are connected with each other by one step of mutation. The method in this paper also works for (totally sign-skew-symmetric) cluster algebras. Moreover, we show that $\mathcal{U}(Σ)$ is invariant under seed mutations when each exchange polynomial…
We introduce a new function on the set of pairs of cluster variables via $f$-vectors, which we call it the compatibility degree (of cluster complexes). The compatibility degree is a natural generalization of the classical compatibility degree introduced by Fomin and Zelevinsky. In particular, we prove that the compatibility degree has the duality property, the symmetry property, the embedding property and the compatibility property, which the classical one has. We also conjecture that the compatibility degree has the exchangeability property. As pieces of evidence of this conjecture, we establ…
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