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Raimund Preusser

Publications and source records attributed to Raimund Preusser.

27 records · Page 2Linked to original sources
arXiv2018

Locally finite weighted Leavitt path algebras

A group graded $K$-algebra $A=\bigoplus\limits_{g\in G} A_g$ is called "locally finite" if $\dim_K A_g < \infty$ for every $g\in G$. We characterise the weighted graphs $(E,w)$ for which the weighted Leavitt path algebra $L_K(E,w)$ is locally finite with respect to its standard grading. We also prove that the locally finite weighted Leavitt path algebras are precisely the Noetherian ones and that $L_K(E,w)$ is locally finite iff $(E,w)$ is finite and the Gelfand-Kirillov dimension of $L_K(E,w)$ equals $0$ or $1$. Further it is shown that a locally finite weighted Leavitt path algebra is isomorphic to a locally finite Leavitt path algebra and therefore is isomorphic to a finite direct sum of matrix algebras over $K$ and $K[X,X^{-1}]$.

math.RA↗
arXiv2018

Weighted Leavitt path algebras of finite Gelfand-Kirillov dimension

We determine the Gelfand-Kirillov dimension of a weighted Leavitt path algebra $L_K(E,w)$ where $K$ is a field and $(E,w)$ a finite weighted graph. Further we show that a finite-dimensional weighted Leavitt path algebra over a field $K$ is isomorphic to a finite product of matrix rings over $K$.

math.RA↗
arXiv2018

Sandwich classification for $O_{2n+1}(R)$ and $U_{2n+1}(R,Δ)$ revisited

In a recent paper, the author proved that if $n\geq 3$ is a natural number, $R$ a commutative ring and $σ\in GL_n(R)$, then $t_{kl}(σ_{ij})$ where $i\neq j$ and $k\neq l$ can be expressed as a product of $8$ matrices of the form $^εσ^{\pm 1}$ where $ε\in E_n(R)$. In this article we prove similar results for the odd-dimensional orthogonal groups $O_{2n+1}(R)$ and the odd-dimensional unitary groups $U_{2n+1}(R,Δ)$ under the assumption that $R$ is commutative and $n\geq 3$. This yields new, short proofs of the Sandwich Classification Theorems for the groups $O_{2n+1}(R)$ and $U_{2n+1}(R,Δ)$.

math.KT↗
arXiv2017

The E-normal structure of odd dimensional unitary groups

In this paper we define odd dimensional unitary groups $U_{2n+1}(R,Δ)$. These groups contain as special cases the odd dimensional general linear groups $GL_{2n+1}(R)$ where $R$ is any ring, the odd dimensional orthogonal and symplectic groups $O_{2n+1}(R)$ and $Sp_{2n+1}(R)$ where $R$ is any commutative ring and further the first author's even dimensional unitary groups $U_{2n}(R,Λ)$ where $(R,Λ)$ is any form ring. We classify the E-normal subgroups of the groups $U_{2n+1}(R,Δ)$ (i.e. the subgroups which are normalized by the elementary subgroup $EU_{2n+1}(R,Δ)$), under the condition that $R$ is either a semilocal or quasifinite ring with involution and $n\geq 3$. Further we investigate the action of $U_{2n+1}(R,Δ)$ by conjugation on the set of all E-normal subgroups.

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arXiv2017

Sandwich classification for $GL_n(R)$, $O_{2n}(R)$ and $U_{2n}(R,Λ)$ revisited

Let $n$ be a natural number greater or equal to $3$, $R$ a commutative ring and $σ\in GL_n(R)$. We show that $t_{kl}(σ_{ij})$ (resp. $t_{kl}(σ_{ii}-σ_{jj}))$ where $i\neq j$ and $k\neq l$ can be expressed as a product of $8$ (resp. $24$) matrices of the form $^εσ^{\pm 1}$ where $ε\in E_n(R)$. We prove similar results for the orthogonal groups $O_{2n}(R)$ and the hyperbolic unitary groups $U_{2n}(R,Λ)$ under the assumption that $R$ is commutative and $n\geq 3$. This yields new, very short proofs of the Sandwich Classification Theorems for the groups $GL_n(R)$, $O_{2n}(R)$ and $U_{2n}(R,Λ)$.

math.KT↗
arXiv2017

Structure of Hyperbolic Unitary Groups II: Classification of E-normal Subgroups

This paper proves the sandwich classification conjecture for subgroups of an even dimensional hyperbolic unitary group $U_{2n}(R,Λ)$ which are normalized by the elementary subgroup $EU_{2n}(R,Λ)$, under the condition that $R$ is a quasi-finite ring with involution, i.e a direct limit of module finite rings with involution, and $n\geq 3$.

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arXiv2017

Applications of normal forms for weighted Leavitt path algebras: simple rings and domains

Weighted Leavitt path algebras (wLpas) are a generalisation of Leavitt path algebras (with graphs of weight 1) and cover the algebras $L_K(n, n + k)$ constructed by Leavitt. Using Bergman's Diamond lemma, we give normal forms for elements of a weighted Leavitt path algebra. This allows us to produce a basis for a wLpa. Using the normal form we classify the wLpas which are domains, simple and graded simple rings. For a large class of weighted Leavitt path algebras we establish a local valuation and as a consequence we prove that these algebras are prime, semiprimitive and nonsingular but contrary to Leavitt path algebras, they are not graded von Neumann regular.

math.RA↗
arXiv2015

Global Actions

A global action is an algebraic analogue of a topological space. It consists of group actions $G_α\curvearrowright X_α$, $(α\inΦ)$, which fulfill a certain compatibility condition. We investigate the homotopy theory of global actions. The main result establishes a Galois type correspondence between connected coverings of a given connected global action and subgroups of the fundamental group of that action.

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