arXiv subjects

Raimund Preusser

Publications and source records attributed to Raimund Preusser.

At least 19 recordsLinked to original sources

arXiv2026

The general linear groupoid of a Leavitt path algebra

The general linear groupoid $\mathbf{G}(L(E))$ of a Leavitt path algebra $L(E)$ is isomorphic to the groupoid whose objects are all $L(E)$-modules of the form $\bigoplus_{i=1}^n v_iL(E)$ where each $v_i$ is a vertex, and whose morphisms are all isomorphisms between these modules. We find a generating set for $\mathbf{G}(L(E))$ and consequently obtain generating sets for all general linear groups $\text{GL}_n(L(E))$ over $L(E)$ (including the group $\text{GL}_1(L(E))$ of invertible elements of $L(E)$). We prove similar results for Leavitt path algebras of hypergraphs (which generalise the Leavitt path algebras of separated graphs and vertex-weighted graphs).

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arXiv2026

On the homogeneous zero components of Leavitt algebras

We prove that the zero component $L(m,n)_0$ of a Leavitt algebra $L(m,n)$ with respect to the canonical grading is a direct limit $\varinjlim_{z}L(m,n)_{0,z}$, where each algebra $L(m,n)_{0,z}$ is a free product of two Bergman algebras. For the special case $m=1,n>1$, one recovers the known result that the zero component $L(1,n)_0$ is a direct limit of matrix algebras. Moreover, we show that $L(m,n)_0$ has the IBN property.

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arXiv2024

Moves for Bergman algebras

We define Bergman presentations and Bergman algebras associated to Bergman presentations. These algebras embrace various generalisations of Leavitt path algebras. A Bergman presentation can be visualised by a Bergman graph, which is a finite bicoloured hypergraph satisfying two conditions. We define several moves for Bergman graphs and prove that they preserve the isomorphism class (respectively the Morita equivalence class) of the corresponding Bergman algebra. One recovers the well-known results, that in the context of finite directed graphs the shift move, outsplitting, insplitting, source elimination and collapsing preserve the isomorphism class (respectively the Morita equivalence class) of the corresponding Leavitt path algebra. Moreover, we mention some connections between Tietze transformations and the moves for Bergman graphs defined in this paper.

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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 $\Gamma$-monoids (monoids with an action of an abelian group $\Gamma$ 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 $\Gamma$, a $\Gamma$-graded ring $R$, and non-zero graded finitely generated projective (left) $R$-modules $P$ and $Q$, we construct a universal $\Gamma$-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 $\Gamma$-monoid $M$, we construct a $\Gamma$-graded ring $S$ such that $\mathcal V^{gr}(S)$ is $\Gamma$-isomorphic to $M$. In fact we show that any finitely generated $\Gamma$-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 $\Gamma$ 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).

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arXiv2023

Modules for Leavitt path algebras of bi-separated graphs via representations graphs

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 for the Leavitt path algebra $L(\dot E)$ of a finitely bi-separated graph $\dot{E}=(E,C,D)$ by introducing the notion of a representation graph for $\dot{E}$. Among these modules we find a class of simple modules. If the bi-separation on $E$ is the Cuntz-Krieger bi-separation (and hence $L(\dot{E})$ is isomorphic to the usual Leavitt path algebra $L(E)$), one recovers the celebrated Chen simple modules.

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arXiv2022

Modules for Leavitt path algebras via extended algebraic branching systems

For a graph $E$, we introduce the notion of an extended $E$-algebraic branching system, generalising the notion of an $E$-algebraic branching system introduced by Gonçalves and Royer. We classify the extended $E$-algebraic branching systems and show that they induce modules for the corresponding Leavitt path algebra $L(E)$. Among these modules we find a class of nonsimple modules whose endomorphism rings are fields.

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arXiv2021

Weighted Leavitt path algebras -- an overview

Weighted Leavitt path algebras were introduced in 2013 by Roozbeh Hazrat. These algebras generalise simultaneously the usual Leavitt path algebras and William Leavitt's algebras $L(m,n)$. In this paper we try to give an overview of what is known about the weighted Leavitt path algebras. We also prove some new results (in particular on the graded K-theory of weighted Leavitt path algebras) and mention open problems.

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arXiv2021

Simple modules for Kumjian-Pask algebras

The paper introduces the notion of a representation $k$-graph $(Δ,α)$ for a given $k$-graph $Λ$. It is shown that any representation $k$-graph for $Λ$ yields a module for the Kumjian-Pask algebra $KP(Λ)$, and the representation $k$-graphs yielding simple modules are characterised. Moreover, the category $RG(Λ)$ of representation $k$-graphs for $Λ$ is investigated using the covering theory of higher-rank graphs.

math.RT↗

arXiv2021

Elementary covering numbers in odd-dimensional unitary groups

Let $(K,Δ)$ be a Hermitian form field and $n\geq 3$. We prove that if $σ\in U_{2n+1}(K,Δ)$ is a unitary matrix of level $(K,Δ)$, then any short root transvection $T_{ij}(x)$ is a product of $4$ elementary unitary conjugates of $σ$ and $σ^{-1}$. Moreover, the bound $4$ is sharp. We also show that any extra short root transvection $T_i(x,y)$ is a product of $12$ elementary unitary conjugates of $σ$ and $σ^{-1}$. If the level of $σ$ is $(0,K\times 0)$, then any $(0,K\times 0)$-elementary extra short root transvection $T_i(x,0)$ is a product of $2$ elementary unitary conjugates of $σ$ and $σ^{-1}$.

math.GR↗

arXiv2021

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 $V_S$. Specialising to graphs with one vertex and $m$ loops of weight $n$, we construct irreducible representations for the celebrated Leavitt algebras $L_K(n,m)$. On the other hand, specialising to graphs with weight one, we recover the simple modules of Leavitt path algebras constructed by Chen via infinite paths or sinks and give a large class of non-simple indecomposable modules.

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arXiv2020

On products of conjugacy classes in general linear groups

Let $K$ be a field and $n\geq 3$. Let $E_n(K)\leq H\leq GL_n(K)$ be an intermediate group and $C$ a noncentral $H$-class. Define $m(C)$ as the minimal positive integer $m$ such that $\exists i_1,\dots,i_m\in\{\pm 1\}$ such that the product $C^{i_1}\dots C^{i_m}$ contains all nontrivial elementary transvections. In this article we obtain a sharp upper bound for $m(C)$. Moreover, we determine $m(C)$ for any noncentral $H$-class $C$ under the assumption that $K$ is algebraically closed or $n=3$ or $n=\infty$.

math.KT↗

arXiv2020

The subnormal structure of classical-like groups over commutative rings

Let $n$ be an integer greater than or equal to $3$ and $(R,Δ)$ a Hermitian form ring where $R$ is commutative. We prove that if $H$ is a subgroup of the odd-dimensional unitary group $U_{2n+1}(R,Δ)$ normalised by a relative elementary subgroup $EU_{2n+1}((R,Δ),(I,Ω))$, then there is an odd form ideal $(J,Σ)$ such that $EU_{2n+1}((R,Δ),(JI^{k},Ω_{\min}^{JI^k}\overset{\cdot}{+}Σ\circ I^{k}))\leq H \leq CU_{2n+1}((R,Δ),(J,Σ))$ where $k=12$ if $n=3$ respectively $k=10$ if $n\geq 4$. As a conseqence of this result we obtain a sandwich theorem for subnormal subgroups of odd-dimensional unitary groups.

math.KT↗

arXiv2019

On general linear groups over exchange rings

Let $R$ be an exchange ring. We prove that the relative elementary subgroups $E_n(R,I)$ are normal in the general linear group $GL_n(R)$ if $n\geq 1$ and that the standard commutator formula $E_n(R,I)=[E_n(R),E_n(R,I)]=[E_n(R),C_n(R,I)]$ holds if $n\geq 3$. Moreover, we classify the subgroups of $GL_n(R)$ that are normalised by the elementary subgroup $E_n(R)$ in the case $n\geq 3$.

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arXiv2019

Reverse decomposition of unipotents over noncommutative rings I: General linear groups

Recently is has been proved that if $σ\in GL_n(R)$ where $R$ is an commutative ring and $n\geq 3$, then each of the elementary transvections $t_{kl}(σ_{ij})~(i\neq j,k\neq l)$ is a product of eight $E_n(R)$-conjugates of $σ$ and $σ^{-1}$. In this article we show that similar results hold true if $R$ is a (noncommutative) von Neumann regular ring, or a Banach algebra, or a ring satisfying a stable range condition, or a ring with Euclidean algorithm, or an almost commutative ring.

math.RA↗

arXiv2019

Weighted Leavitt path algebras that are isomorphic to unweighted Leavitt path algebras

Let $K$ be a field. We characterise the row-finite weighted graphs $(E,w)$ such that the weighted Leavitt path algebra $L_K(E,w)$ is isomorphic to an unweighted Leavitt path algebra. Moreover, we prove that if $L_K(E,w)$ is locally finite, or Noetherian, or Artinian, or von Neumann regular, or has finite Gelfand-Kirillov dimension, then $L_K(E,w)$ is isomorphic to an unweighted Leavitt path algebra.

math.RA↗

arXiv2019

Leavitt path algebras of hypergraphs

We define Leavitt path algebras of hypergraphs generalizing simultaneously Leavitt path algebras of finitely separated graphs and Leavitt path algebras of row-finite vertex-weighted graphs. We find linear bases for those algebras, compute their Gelfand-Kirillov dimension, obtain some results on ring-theoretic properties like simplicity, von Neumann regularity and Noetherianess and investigate their K-theory and graded K-theory. By doing so we obtain new results on the Gelfand-Kirillov dimension and graded K-theory of Leavitt path algebras of separated graphs and on the graded K-theory of weighted Leavitt path algebras.

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arXiv2018

The E-normal structure of Petrov's odd unitary groups over commutative rings

For an odd quadratic space $V$ of Witt index $\geq 3$ over a commutative ring with pseudoinvolution, we classify the subgroups of the odd unitary group $U(V)$ that are normalized by the elementary subgroup $EU_{(e_1,e_{-1})}(V)$ defined by a hyperbolic pair $(e_1,e_{-1})$ in $V$. Further we correct some minor mistakes that exist in the literature on odd unitary groups.

math.KT↗

arXiv2018

The V-monoid of a weighted Leavitt path algebra

We compute the $V$-monoid of a weighted Leavitt path algebra of a row-finite weighted graph, correcting a wrong computation of the $V$-monoid that exists in the literature. Further we show that the description of $K_0$ of a weighted Leavitt path algebra that exists in the literature is correct (although the computation was based on a wrong $V$-monoid description).

math.RA↗