The general linear groupoid of a Leavitt path algebra

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 ${⨁}_{i=1}^n{v}_{i}L(E)$ 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).