arXiv subjects

Roozbeh Hazrat

Publications and source records attributed to Roozbeh Hazrat.

At least 19 recordsLinked to original sources

arXiv2026

Some more talents of the talented monoid of a higher-rank graph

In this paper, we explore the idea that the graded Grothendieck group $K_0^{gr}$, or equivalently its positive cone, the talented monoid, can detect the structural type of higher-rank graph algebras (i.e., higher-rank graph $C^*$-algebras and Kumjian--Pask algebras). We show that the talented monoid captures some of the essential geometric information of a higher-rank graph, including the existence of cycles with and without entrances. In turn, we show that the graded $K$-theory can effectively distinguish the class of locally finite Kumjian--Pask algebras, and also the class of crossed product Kumjian--Pask algebras. We also derive talented monoid criteria for higher-rank graph algebras to be purely infinite simple, and not to be $AF$ or ultramatricial.

math.RA↗

arXiv2026

Higher-Dimensional Symbolic Dynamics: A Textile Framework For 3-graphs

Textile systems are best known to model two-dimensional shifts of finite type. In this article, we associate a discrete algebra with a textile system and provide a groupoid model for it. When the textile system is left-resolving, this algebra coincides with the Kumjian--Pask algebra of the associated $2$-graph. The main objective of this paper is to extend textile systems to dimension $3$ so that the resulting structures can, on the one hand, capture all three-dimensional shifts of finite type and, on the other hand, provide a textile-like framework for $3$-graphs extending the well-known connection between $2$-graphs and left-resolving textile systems. We introduce a model of a three-dimensional textile system and investigate the interplay between such textile systems and $3$-graphs. In particular, our investigation shows that the conditions required to form a $3$-graph from a $3$-colored graph (including the delicate associativity condition on tricolored paths), can be encoded in terms of simple pullback diagrams arising from the textile data. We also define homology groups for three-dimensional textiles and prove that these groups coincide with the homology groups of the associated $3$-graphs, thus establishing that our construction is homologically consistent with $3$-graphs.

math.DS↗

arXiv2026

The graded Grothendieck group ${K}_0^{\mathrm{gr}}$ is full for weighted Leavitt path algebras

The Graded Classification Conjecture asserts that the graded Grothendieck group $K_0^{\mathrm{gr}}$ is a complete invariant for the classes of Leavitt path algebras and graph $C^*$-algebras. The conjecture remains open, as neither a proof nor a counterexample is currently known. In this article, we extend the study of this invariant to the class of vertex-weighted Leavitt path algebras. We show that $K_0^{\mathrm{gr}}$ distinguishes weighted Leavitt path algebras from ordinary (unweighted) Leavitt path algebras. We further prove that an isomorphism between the graded Grothendieck groups of weighted Leavitt path algebras induces an isomorphism between the corresponding semilattices of vertex-generated ideals. In addition, we show that $K_0^{\mathrm{gr}}$ classifies the classical Leavitt algebras $L_K(n,n+k)$. Next, we prove that $K_0^{\mathrm{gr}}$ is a full functor on the category of all weighted Leavitt path algebras. Consequently, in the special case where all weights are equal to $1$, we recover the lifting theorem established independently by Arnone and Vas for Leavitt path algebras. This confirms one direction of the Graded Classification Conjecture for Leavitt path algebras.

math.RA↗

arXiv2025

Embedding $K$-algebras into Leavitt algebra $L_K(1, 2)$

Since the commutative monoid $T = (\{0, 1\}, \vee)$ is a weak terminal object in the category of conical monoids with order units, there is a unital homomorphism from every Bergman $K$-algebra corresponding to a conical finitely generated commutative monoid into the Leavitt algebra $L_K(1,2)$, where $K$ is a field. This fact will be used to give a short proof that Leavitt path algebras associated with finite graphs with condition $(L)$ embed into $L_K(1,2)$, as well as provide criteria for an embedding of $M_s(L_{K}(1, m))$ in $M_s(L_{K}(1, n))$. As our second main result, we show that the Heisenberg equation $xy-yx=1$ cannot be realized in any Steinberg algebra, implying that the first Weyl algebra cannot be embedded into $L_K(1,2)$, giving an affirmative answer to a question of Brownlowe and Sorensen on the embeddability of $K$-algebras with a countable basis inside $L_K(1,2)$. Whereas, $L_K(E)$ cannot be graded-embedded into $L_K(1,2)$ in general, in the final section we show that $L_K(E)$ does admit a graded embedding into $L_K(1,2)\otimes_K L_K(1,2)$.

math.RA↗

arXiv2025

Graded Lawson-Stone Duality

The classical Stone duality associates to each Boolean algebra a topological space consisting of ultrafilters. Lawson's generalisation constructs a dual equivalence of categories of Boolean inverse $\land$-semigroups and Hausdorff ample topological groupoids. We further generalise the duality to graded versions of those categories, while allowing larger classes of morphisms, and provide various illustrations.

math.RA↗

arXiv2025

Higher-rank graphs and the graded $K$-theory of Kumjian-Pask algebras

This paper lays out the foundations of graded $K$-theory for Leavitt algebras associated with higher-rank graphs, also known as Kumjian-Pask algebras, establishing it as a potential tool for their classification. For a row-finite $k$-graph $\Lambda$ without sources, we show that there exists a $\mathbb{Z}[\mathbb{Z}^k]$-module isomorphism between the graded zeroth (integral) homology $H_0^{gr}(\mathcal{G}_\Lambda)$ of the infinite path groupoid $\mathcal{G}_\Lambda$ and the graded Grothendieck group $K_0^{gr}(KP_\mathsf{k}(\Lambda))$ of the Kumjian-Pask algebra $KP_\mathsf{k}(\Lambda)$, which respects the positive cones (i.e., the talented monoids). We demonstrate that the $k$-graph moves of in-splitting and sink deletion defined by Eckhardt et al. (Canad. J. Math. 2022) preserve the graded $K$-theory of associated Kumjian-Pask algebras and produce algebras which are graded Morita equivalent, thus providing evidence that graded $K$-theory may be an effective invariant for classifying certain Kumjian-Pask algebras. We also determine a natural sufficient condition regarding the fullness of the graded Grothendieck group functor. More precisely, for two row-finite $k$-graphs $\Lambda$ and $\Omega$ without sources and with finite object sets, we obtain a sufficient criterion for lifting a pointed order-preserving $\mathbb{Z}[\mathbb{Z}^k]$-module homomorphism between $K_0^{gr}(KP_\mathsf{k}(\Lambda))$ and $K_0^{gr}(KP_\mathsf{k}(\Omega))$ to a unital graded ring homomorphism between $KP_\mathsf{k}(\Lambda)$ and $KP_\mathsf{k}(\Omega)$. For this we adopt, in the setting of $k$-graphs, the bridging bimodule technique recently introduced by Abrams, Ruiz and Tomforde (Algebr. Represent. Theory 2024).

math.KT↗

arXiv2025

Williams' conjecture holds for graphs of Gelfand-Kirillov dimension three

A graph of Gelfand-Kirillov dimension three is a connected finite essential graph such that its Leavitt path algebra has Gelfand-Kirillov dimension three. We provide number-theoretic criteria for graphs of Gelfand-Kirillov dimension three to be strong shift equivalent. We then prove that two graphs of Gelfand-Kirillov dimension three are shift equivalent if and only if they are strongly shift equivalent, if and only if their corresponding Leavitt path algebras are graded Morita equivalent, if and only if their graded $K$-theories, $K^{\text{gr}}_0$, are order-preserving $\mathbb{Z}[x, x^{-1}]$-module isomorphic. As a consequence, we obtain that the Leavitt path algebras of graphs of Gelfand-Kirillov dimension three are graded Morita equivalent if and only if their graph $C^*$-algebras are equivariant Morita equivalent, and two graphs $E$ and $F$ of Gelfand-Kirillov dimension three are shift equivalent if and only if the singularity categories $\text{D}_{\text{sg}}(KE/J_E^2)$ and $\text{D}_{\text{sg}}(KF/J_F^2)$ are triangulated equivalent.

math.RA↗

arXiv2025

The uniform dimension of a monoid with applications to graph algebras

We adapt Goldie's concept of uniform dimensions from module theory over rings to $\Gamma$-monoids. A $\Gamma$-monoid $M$ is said to have uniform dimension $n$ if $n$ is the largest number of pairwise incomparable nonzero $\Gamma$-order ideals contained in $M$. Specializing to the talented monoid of a graph, we show that the uniform dimension provides a rough measure of how the graph branches out. Since for any order ideal $I$, its orthogonal ideal $I^\perp$ is the largest ideal incomparable to $I$, we study the notions of orthogonality and regularity, particularly when $I^{\perp\perp}=I$. We show that the freeness of the action of $\mathbb Z$ on the talented monoid of a graph is preserved under quotienting by a regular ideal. Furthermore, we determine the underlying hereditary and saturated sets that generate these ideals. These results unify recent studies on regular ideals of the corresponding Leavitt path algebras and graph $C^*$-algebras. We conclude that for graphs $E$ and $F$, if there is a $\mathbb Z$-monoid isomorphism $T_E\cong T_F$, then there is a one-to-one correspondence between the regular ideals of the associated Leavitt path algebras $L_K(E)$ and $L_K(F)$ (and similarly, $C^*(E)$ and $C^*(F)$). Since the talented monoid $T_E$ is the positive cone of the graded Grothendieck group $K_0^{gr}(L_K(E))$, this provides further evidence supporting the Graded Classification Conjecture.

math.RA↗

arXiv2024

On structural connections between sandpile monoids and weighted Leavitt path 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 a sandpile graph $E$ is both isomorphic to the lattice of all nonempty saturated hereditary subsets of $E$, the lattice of all order-ideals of $\text{SP}(E)$ and the lattice of all ideals of the weighted Leavitt path algebra $L_{K}(E, \omega)$ generated by vertices. Also, we describe the sandpile group of a sandpile graph $E$ via archimedean classes of $\text{SP}(E)$, and prove that all maximal subgroups of $\text{SP}(E)$ are exactly the Grothendieck groups of these archimedean classes. Finally, we give the structure of the Leavitt path algebra $L_{K}(E)$ of a sandpile graph $E$ via a finite chain of graded ideals being invariant under every graded automorphism of $L_{K}(E)$, and completely describe the structure of $L_{K}(E)$ such that the lattice of all idempotents of $\text{SP}(E)$ is a chain. Consequently, we completely describe the structure of the weighted Leavitt path algebra of a sandpile graph $E$ such that $\text{SP}(E)$ has exactly two idempotents.

math.RA↗

arXiv2024

The Talented Monoid of Higher-Rank Graphs with Applications to Kumjian-Pask Algebras

Given a row-finite higher-rank $k$-graph $\Lambda$, we define a commutative monoid $T_\Lambda$ which is a higher-rank analogue of the talented monoid of a directed graph. The talented monoid $T_\Lambda$ is canonically a $\mathbb{Z}^k$-monoid with respect to the action of state shift. This monoid coincides with the positive cone of the graded Grothendieck group $K_0^{gr}(KP_\mathsf{k}(\Lambda))$ of the Kumjian-Pask algebra $KP_\mathsf{k}(\Lambda)$ with coefficients in a field $\mathsf{k}$. The aim of the paper is to investigate this $\mathbb{Z}^k$-monoid as a capable invariant for classification of Kumjian-Pask algebras. If $\mathbb{Z}^k$ acts freely on $T_\Lambda$ (i.e., if $T_\Lambda$ has no nonzero periodic element), then we show that the $k$-graph $\Lambda$ is aperiodic. The converse is also proved to be true provided $\Lambda$ has no sources and $T_\Lambda$ is atomic. Moreover in this case, we provide a talented monoid characterization for strongly aperiodic $k$-graphs. We prove that for a row-finite $k$-graph $\Lambda$ without sources, cofinality is equivalent to the simplicity of $T_\Lambda$ as a $\mathbb{Z}^k$-monoid. In view of this we provide a talented monoid criterion for the Kumjian-Pask algebra $KP_R(\Lambda)$ of $\Lambda$ over a unital commutative ring $R$ to be graded basic ideal simple. We also describe the minimal left ideals of $KP_\mathsf{k}(\Lambda)$ in terms of the aperiodic atoms of $T_\Lambda$ and thus obtain a monoid theoretic characterization for $Soc(KP_\mathsf{k}(\Lambda)$) to be an essential ideal. These results help us to characterize semisimple Kumjian-Pask algebras through the lens of $T_\Lambda$.

math.RA↗

arXiv2024

Williams' and Graded Equivalence Conjectures for small graphs

We prove what might have been expected: The Williams Conjecture in symbolic dynamics and Graded Morita Equivalence Conjecture for Leavitt/$C^*$-graph algebras hold for ``small graphs'', i.e., connected graphs with three vertices, no parallel edges, no sinks with no trivial hereditary and saturated subsets. Namely, two small graphs are shift equivalent if and only if they are strong shift equivalent if and only if their Leavitt/$C^*$-graph algebras are graded/equivariant Morita equivalent.

math.RA↗

arXiv2024

Morita theory of finite representations of Leavitt path algebras

The Graded Classification Conjecture states that for finite directed graphs $E$ and $F$, the associated Leavitt path algebras $L_\K(E)$ and $L_\K(F)$ are graded Morita equivalent, i.e., $\Gr L_\K(E) \approx_{\gr} \Gr L_\K(F)$, if and only if, their graded Grothendieck groups are isomorphic $K_0^{\gr}(L_\K(E)) \cong K_0^{\gr}(L_\K(F))$ as order-preserving $\mathbb Z[x,x^{-1}]$-modules. Furthermore, if under this isomorphism, the class $[L_\K(E)]$ is sent to $[L_\K(F)]$ then the algebras are graded isomorphic, i.e., $L_\K(E) \cong _{\gr} L_\K(F)$. In this note we show that, for finite graphs $E$ and $F$ with so sinks and sources, an order-preserving $\mathbb Z[x,x^{-1}]$-module isomorphism $K_0^{\gr}(L_\K(E)) \cong K_0^{\gr}(L_\K(F))$ gives that the categories of locally finite dimensional graded modules of $L_\K(E)$ and $L_\K(F)$ are equivalent, i.e., $\fGr[\mathbb{Z}] L_\K(E)\approx_{\gr} \fGr[\mathbb{Z}]L_\K(F).$ We further obtain that the category of finite dimensional (graded) modules are equivalent, i.e., $\fModd L_\K(E) \approx \fModd L_\K(F)$ and $\fGr L_\K(E) \approx_{\gr} \fGr L_\K(F)$.

math.RT↗

arXiv2024

Unital aligned shift equivalence and the graded classification conjecture for Leavitt path algebras

We prove that a unital shift equivalence induces a graded isomorphism of Leavitt path algebras when the shift equivalence satisfies an alignment condition. This yields another step towards confirming the Graded Classification Conjecture. Our proof uses the bridging bimodule developed by Abrams, the fourth-named author and Tomforde, as well as a general lifting result for graded rings that we establish here. This general result also allows us to provide simplified proofs of two important recent results: one independently proven by Arnone and Va{\v s} through other means that the graded $K$-theory functor is full, and the other proven by Arnone and Cortiñas that there is no unital graded homomorphism between a Leavitt algebra and the path algebra of a Cuntz splice.

math.RA↗

arXiv2024

Monoids, dynamics and Leavitt path algebras

Leavitt path algebras, which are algebras associated to directed graphs, were first introduced about 20 years ago. They have strong connections to such topics as symbolic dynamics, operator algebras, non-commutative geometry, representation theory, and even chip firing. In this article we invite the reader to sneak a peek at these fascinating algebras and their interplay with several seemingly disparate parts of mathematics.

math.RA↗

arXiv2024

Bergman algebras: The graded universal algebra constructions

A half a century ago, George Bergman introduced stunning machinery which would realise any commutative conical monoid as the non-stable $K$-theory of a ring. The ring constructed is ``minimal" or ``universal". Given the success of graded $K$-theory in classification of algebras and its connections to dynamics and operator algebras, the realisation of $Γ$-monoids (monoids with an action of an abelian group $Γ$ on them) as non-stable graded $K$-theory of graded rings becomes vital. In this paper, we revisit Bergman's work and develop the graded version of this universal construction. For an abelian group $Γ$, a $Γ$-graded ring $R$, and non-zero graded finitely generated projective (left) $R$-modules $P$ and $Q$, we construct a universal $Γ$-graded ring extension $S$ such that $S\otimes_R P\cong S\otimes_R Q$ as graded $S$-modules. This makes it possible to bring the graded techniques, such as smash products and Zhang twists into Bergman's machinery. Given a commutative conical $Γ$-monoid $M$, we construct a $Γ$-graded ring $S$ such that $\mathcal V^{gr}(S)$ is $Γ$-isomorphic to $M$. In fact we show that any finitely generated $Γ$-monoid can be realised as the non-stable graded $K$-theory of a hyper Leavitt path algebra. Here $\mathcal V^{gr}(S)$ is the monoid of isomorphism classes of graded finitely generated projective $S$-modules and the action of $Γ$ on $\mathcal V^{gr}(S)$ is by shift of degrees. Thus the group completion of $M$ can be realised as the graded Grothendieck group $K^{\gr}_0(S)$. We use this machinery to provide a short proof to the fullness of the graded Grothendieck functor $K^{gr}_0$ for the class of Leavitt path algebras (i.e., Graded Classification Conjecture II).

math.RA↗

arXiv2024

Classification conjectures for Leavitt path algebras

The theory of Leavitt path algebras is intrinsically related, via graphs, to the theory of symbolic dynamics and $C^*$-algebras where the major classification programs have been a domain of intense research in the last 50 years. In this survey article, we gather together current lines of research in the classification of Leavitt path algebras, questions, conjectures, and some of the results about them that have been obtained so far.

math.RA↗

arXiv2023

Unital algebras being Morita equivalent to weighted Leavitt path algebras

In this article, we describe the endomorphism ring of a finitely generated progenerator module of a weighted Leavitt path algebra $L_{K}(E, w)$ of a finite vertex weighted graph $(E, w)$. Contrary to the case of Leavitt path algebras, we show that a (full) corner of a weighted Leavitt path algebra is, in general, not isomorphic to a weighted Leavitt path algebra. However, using the above result, we show that for every full idempotent $ε$ in $L_{K}(E, w)$, there exists a positive integer $n$ such that $M_n(εL_{K}(E, w) ε)$ is isomorphic to the weighted Leavitt path algebra of a weighted graph explicitly constructed from $(E, w)$. We then completely describe unital algebras being Morita equivalent to weighted Leavitt path algebras of vertex weighted graphs. In particular, we characterize unital algebras being Morita equivalent to sandpile algebras.

math.RA↗

arXiv2023

Comparability in the graph monoid

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 features with the graph-theoretic properties of the generators of the element. We also characterize graphs such that every element of $M_E^Γ$ is comparable, periodic, graphs such that every nonzero element of $M_E^Γ$ is aperiodic, incomparable, graphs such that no nonzero element of $M_E^Γ$ is periodic, and graphs such that no element of $M_E^Γ$ is aperiodic. The Graded Classification Conjecture can be formulated to state that $M_E^Γ$ is a complete invariant of the Leavitt path algebra $L_K(E)$ of $E$ over a field $K.$ Our characterizations indicate that the Graded Classification Conjecture may have a positive answer since the properties of $E$ are well reflected by the structure of $M_E^Γ.$ Our work also implies that some results of [R. Hazrat, H. Li, The talented monoid of a Leavitt path algebra, J. Algebra, 547 (2020) 430-455] hold without requiring the graph to be row-finite.

math.RA↗