Related papers: Two-Sided Ideals in Leavitt Path Algebras
In this article, we characterize the class of complementary edge ideals which satisfy the licci property in terms of the underlying graph. Using this characterization, we associate the licci property of a complementary edge ideal to its…
We apply some basic notions from combinatorial topology to establish various algebraic properties of edge ideals of graphs and more general Stanley-Reisner rings. In this way we provide new short proofs of some theorems from the literature…
For a field K and directed graph E, we analyze those elements of the Leavitt path algebra L_K(E) which lie in the commutator subspace [L_K(E), L_K(E)]. This analysis allows us to give easily computable necessary and sufficient conditions to…
A biased graph consists of a graph $G$ together with a collection of distinguished cycles of $G$, called balanced cycles, with the property that no theta subgraph contains exactly two balanced cycles. Perhaps the most natural biased graphs…
We construct some irreducible representations of the Leavitt path algebra of an arbitrary quiver. The constructed representations are associated to certain algebraic branching systems. For a row-finite quiver, we classify algebraic…
Given a base field $\Bbbk$ of characteristic zero, for each graph $G$, we associate the artinian algebra $A(G)$ defined by the edge ideal of $G$ and the squares of the variables. We study the weak Lefschetz property of $A(G)$. We classify…
In this article, we introduce a relation including ideals of an evolution algebra and hereditary subsets of vertices of its associated graph and establish some properties among them. This relation allows us to determine maximal ideals and…
Leavitt path algebras of bi-separated graphs have been recently introduced by R. Mohan and B. Suhas. These algebras provide a common framework for studying various generalisations of Leavitt path algebras. In this paper we obtain modules…
We study the family of graphs whose number of primitive cycles equals its cycle rank. It is shown that this family is precisely the family of ring graphs. Then we study the complete intersection property of toric ideals of bipartite graphs…
In this article, we provide an explicit description of a set of generators for any ideal of an ultragraph Leavitt path algebra. We provide several additional consequences of this description, including information about generating sets for…
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…
In this article, we describe the relation between the properties of being equational noetherian and ascending chain condition on ideals of an arbitrary algebra. We also give a formulation of Hilbert's basis theorem for varieties of algebras…
In this article, we establish the relations between a sandpile graph, its sandpile monoid and the weighted Leavitt path algebra associated with it. Namely, we show that the lattice of all idempotents of the sandpile monoid $\text{SP}(E)$ of…
We prove that, for the edge ideal of a graph whose cycles are pairwise vertex-disjoint, the arithmetical rank is bounded above by the sum of the number of cycles and the maximum height of its associated primes.
This introduction to graphs and graph algebras provides the optimal bound for the number of all paths of length $k$ in a graph with $N\geq k$ edges and no loops. Our proof relies on a construction of a number of terminating algorithms that…
We characterize the graphs $G$ for which their toric ideals $I_G$ are complete intersections. In particular we prove that for a connected graph $G$ such that $I_G$ is complete intersection all of its blocks are bipartite except of at most…
Over fields of characteristic two, we construct an infinite ascending chain of GL-stable ideals in the coordinate ring of infinite skew-symmetric matrices. This construction provides the first known example of a non-noetherian GL-algebra,…
We investigate the ascending Loewy socle series of Leavitt path algebras $L_K(E)$ for an arbitrary graph $E$ and field $K$. We classify those graphs $E$ for which $L_K(E)=S_{\lambda}$ for some element $S_{\lambda}$ of the Loewy socle…
If $E$ is a not-necessarily row-finite graph, such that each vertex of $E$ emits at most countably many edges, then a {\it desingularization} $F$ of $E$ can be constructed (see e.g. (1) G. Abrams, G. Aranda Pino, Leavitt path algebras of…
In this paper we give a new family of complete intersections which have the strong Lefschetz property. The family consists of (Artinian algebras defined by) ideals generated by power sum symmetric polynomials of consecutive degrees and of…