Leavitt路代数的一般线性群胚
The general linear groupoid of a Leavitt path algebra
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中文总结 AI 辅助
本文研究Leavitt路代数的一般线性群胚，确定其生成集并推导对应一般线性群的生成集，还将结果推广到超图的Leavitt路代数。
中文摘要 AI 辅助
Leavitt路代数$L(E)$的一般线性群胚$\boldsymbol{G}(L(E))$同构于以形如$\bigoplus_{i=1}^n v_iL(E)$的所有$L(E)$-模为对象（其中每个$v_i$是一个顶点）、以这些模之间的所有同构为态射的群胚。我们找到$\boldsymbol{G}(L(E))$的一个生成集，进而得到$L(E)$上所有一般线性群$\text{GL}_n(L(E))$（包括$L(E)$可逆元构成的群$\text{GL}_1(L(E))$）的生成集。我们对超图的Leavitt路代数也证明了类似结果，这类代数推广了分离图和顶点加权图的Leavitt路代数。
英文摘要
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).
