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arXiv · 2608.21026

The general linear groupoid of a Leavitt path algebra

Raimund Preusser
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Abstract

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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Raimund Preusser. 2026-08-21. The general linear groupoid of a Leavitt path algebra. https://arxiv.org/abs/2608.21026

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