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Tran Giang Nam

Publications and source records attributed to Tran Giang Nam.

At least 19 recordsLinked to original sources
arXiv2026

When twist of a Leavitt path algebra is again a Leavitt path algebra

In this paper, we study Zhang twist of Leavitt path algebra $L_K(E)$ using a graded automorphism $σ$ induced by a graph automorphism such that the twisted algebra is a Leavitt path algebra over another graph $E_σ$ which is a twisted graph obtained from the original graph $E$. We establish combinatorial connection between the graphs $E$ and $E_σ$ and as a consequence, we show that many ring-theoretic properties are invariant for Leavitt path algebras under the twist by $σ$. We also define the notion of noncommutative projective scheme for $\mathbb Z$-graded algebras that coincides with the notion of noncommutative projective scheme for connected $\mathbb N$-graded algebras defined by Artin and Zhang and study it in the context of twists of Leavitt path algebras.

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arXiv2026

Dynamics on graphs with disjoint cycles and applications

In this article, we introduce the notion of connected finite graphs with disjoint cycles in normal form and show that any such graph can be transformed into a normal form graph via a finite sequence of in-splittings and out-splittings. Consequently, we provide number-theoretic criteria for meteor graphs of length three to be strongly shift equivalent, where a meteor graph of length three is a connected finite essential graph consisting of three disjoint cycles which makes a unique chain of cycles of length three. We then prove that meteor graphs of length three whose cycle lengths are pairwise coprime 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, $K^{gr}_0$, are order-preserving $\mathbb{Z}[x, x^{-1}]$-module isomorphic. As a consequence, Williams' Conjecture and Hazrat's Graded Morita Equivalence Conjecture hold for graphs with disjoint cycles that contain exactly three cycles whose lengths are pairwise coprime.

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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.

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arXiv2026

The Reduction Theorem for Leavitt Labelled Path Algebras and Its Applications

We introduce a notion of labelled cycle for normal labelled spaces and prove a reduction theorem for Leavitt labelled path algebras. We show that every nonzero element can be reduced, by suitable left and right multiplication, either to a nonzero scalar multiple of a projection or to a polynomial supported on a labelled cycle without exits. This extends the classical reduction theorem for Leavitt path algebras of directed graphs and its analogues for ultragraph Leavitt path algebras and subshift algebras. As applications, we prove the graded uniqueness theorem and the Cuntz--Krieger uniqueness theorem for Leavitt labelled path algebras, and show that these algebras are semiprime and semiprimitive over fields.

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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 $L_K(1,2)\otimes_K L_K(1,2)$.

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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.

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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 $L_{K}(E, ω)$ 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.

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arXiv2024

Automorphisms of Leavitt path algebras: Zhang twist and irreducible representations

In this article, we construct (graded) automorphisms fixing all vertices of Leavitt path algebras of arbitrary graphs in terms of general linear groups over corners of these algebras. As an application, we study Zhang twist of Leavitt path algebras and describe new classes of irreducible representations of Leavitt path algebras of the rose graphs $R_n$ with $n$ petals.

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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.

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arXiv2022

Products of commutator ideals of some Lie-admissible algebras

In this article, we mainly study the products of commutator ideals of Lie-admissible algebras such as Novikov algebras, bicommutative algebras, and assosymmetric algebras. More precisely, we first study the properties of the lower central chains for Novikov algebras and bicommutative algebras. Then we show that for every Lie nilpotent Novikov algebra or Lie nilpotent bicommutative algebra $\mathcal{A}$, the ideal of $\mathcal{A}$ generated by the set $\{ab - ba\mid a, b\in \mathcal{A}\}$ is nilpotent. Finally, we study properties of the lower central chains for assosymmetric algebras, study the products of commutator ideals of assosymmetric algebras and show that the products of commutator ideals have a similar property as that for associative algebras.

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arXiv2021

Congruence-simplicity of Steinberg algebras of non-Hausdorff ample groupoids over semifields

We investigate the algebra of an ample groupoid, introduced by Steinberg, over a semifield S. In particular, we obtain a complete characterization of congruence-simpleness for Steinberg algebras of second-countable ample groupoids, extending the well-known characterizations when S is a field. We apply our congruence-simplicity results to tight groupoids of inverse semigroup representations associated to self-similar graphs.

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arXiv2020

Lie nilpotent Novikov algebras and Lie solvable Leavitt path algebras

In this paper, we first study properties of the lower central chains for Novikov algebras. Then we show that for every Lie nilpotent Novikov algebra~$\mathcal{N}$, the ideal of~$\mathcal{N}$ generated by the set~$\{ab - ba\mid a, b\in \mathcal{N}\}$ is nilpotent. We secondly provide necessary and sufficient conditions on the graph $E$ and the field $K$ for which the Leavitt path algebra $L_K(E)$ is Lie solvable. Consequently, we obtain a complete description of Lie nilpotent Leavitt path algebras, and show that the Lie solvability of~$L_K(E)$ and the Lie nilpotency of $[L_K(E),L_K(E)]$ are the same.

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arXiv2020

On Steinberg algebras of Hausdorff ample groupoids over commutative semirings

We investigate the algebra of a Hausdorff ample groupoid, introduced by Steinberg, over a commutative semiring S. In particular, we obtain a complete characterization of congruence-simpleness for such Steinberg algebras, extending the well-known characterizations when S is a field or a commutative ring. We also provide a criterion for the Steinberg algebra of the graph groupoid associated to an arbitrary graph to be congruence-simple. Motivated by a result of Clark and Sims, we show that, over the Boolean semifield, the natural homomorphism from the Leavitt path algebra to the Steinberg algebra is an isomorphism if and only if the associated graph is row-finite. Moreover, we establish the Reduction Theorem and Uniqueness Theorems for Leavitt path algebras of row-finite graphs over the Boolean semifield.

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arXiv2020

Purely infinite simple ultragraph Leavitt path algebras

In this article, we give necessary and sufficient conditions under which the Leavitt path algebra $L_K(\mathcal{G})$ of an ultragraph $\mathcal{G}$ over a field $K$ is purely infinite simple and that it is von Neumann regular. Consequently, we obtain that every graded simple ultragraph Leavitt path algebra is either a locally matricial algebra, or a full matrix ring over $K[x, x^{-1}]$, or a purely infinite simple algebra.

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arXiv2020

Simple Lie algebras arising from Steinberg algebras of Hausdorff ample groupoids

In this paper, we show that a unital simple Steinberg algebra is central, and a nonunital simple Steinberg algebra has zero center. We identify the fields $K$ and Hausdorff ample groupoids $\mathcal{G}$ for which the simple Steinberg algebra $A_K(\mathcal{G})$ yields a simple Lie algebra $[A_K(\mathcal{G}), A_K(\mathcal{G})]$. We apply the obtained results on simple Leavitt path algebras, simple Kumjian-Pask algebras and simple Exel-Pardo algebras to determine their associated Lie algebras are simple. In particular, we give easily computable criteria to determine which Lie algebras of the form $[L_K(E), L_K(E)]$ are simple, when $E$ is an arbitrary graph and the Leavitt path algebra $L_K(E)$ is simple. Also, we obtain that unital simple Exel-Pardo algebras are central, and nonunital simple Exel-Pardo algebras have zero center.

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arXiv2017

On congruence-semisimple semirings and the $K_0$-group characterization of ultramatricial algebras over semifields

In this paper, we provide a complete description of congruence-semisimple semirings and introduce the pre-ordered abelian Grothendieck groups $K_0(S)$ and $SK_0(S)$ of the isomorphism classes of the finitely generated projective and strongly projective S-semimodules, respectively, over an arbitrary semiring S. We prove that the $SK_0$-groups and $K_0$-groups are complete invariants of, i.e., completely classify, ultramatricial algebras over a semifield F. Consequently, we show that the $SK_0$-groups completely characterize zerosumfree congruence-semisimple semirings.

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arXiv2015

Simpleness of Leavitt Path Algebras with Coefficients in a Commutative Semiring

In this paper, we study ideal- and congruence-simpleness for the Leavitt path algebras of directed graphs with coefficients in a commutative semiring S, as well as establish some fundamental properties of those algebras. We provide a complete characterization of ideal-simple Leavitt path algebras with coefficients in a semifield S that extends the well-known characterizations when the ground semiring S is a field. Also, extending the well-known characterizations when S is a field or commutative ring, we present a complete characterization of congruence-simple Leavitt path algebras over row-finite graphs with coefficients in a commutative semiring S.

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