The injective envelope of simple modules over Leavitt path algebras II: simples arising from exclusive cycles
Year: 2026
Type: preprint
Abstract: Let $K$ be any field, $E$ any directed graph, and $L_K(E)$ the associated Leavitt path algebra. As described first by Chen, and subsequently generalized by Ara and Rangaswamy, for each cycle $c$ in $E$ one can build the simple left $L_K(E)$-module $V_c^E$, and then more generally $V_{p(x),c}^E$ (where $p(x)$ is an irreducible polynomial in $K[x,x^{-1}]$). A cycle $c$ is called {\it exclusive} in case none of the vertices of $c$ is the base of any cycle other than $c$. In our main result we provide an explicit description of the injective envelope of $V_c^E$, and then more generally of $V_{p(x),c}^E$, for each exclusive cycle $c$. Our method involves defining an $L_K(E)$-module structure on an appropriately-built $K$-vector space of infinite series. Our main result significantly generalizes previous work of the authors, in that the result holds for all graphs (finite or not), and all exclusive cycles.
Source: arXiv (Cornell University)
Authors: Abrams, G., Mantese, F., Tonolo, A.
Language: English
Topic: Rings, Modules, and Algebras
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7206151144