Gabriel's Theorem for Infinite Quivers

2023/12/15 by Nathaniel Gallup, Gallup, Nathaniel, Stephen Sawin +1
Mathematics · #Algebraic structures and combinatorial models #Homotopy and Cohomology in Algebraic Topology #Advanced Combinatorial Mathematics

paper · pdf · doi:10.48550/arxiv.2312.09904

Abstract

We prove a version of Gabriel's theorem for (possibly infinite dimensional) representations of infinite quivers. More precisely, we show that the representation theory of quiver Ω is of unique type (each dimension vector has at most one associated indecomposable) and infinite Krull-Schmidt (every, possibly infinite dimensional, representation is a direct sum of indecomposables) if and only if Ω is eventually outward and of generalized ADE Dynkin type (A_n, D_n, E₆, E₇, E₈, A_∞, A_{∞, ∞}, or D_∞). Furthermore we define an analog of the Euler-Tits form on the space of eventually constant infinite roots and show that a quiver is of generalized ADE Dynkin type if and only if this form is positive definite. In this case the indecomposables are all locally finite-dimensional and eventually constant and correspond bijectively to the positive roots (i.e. those of length 1).

Related

• Gabriel's Theorem for Locally Finite-Dimensional Representations of Infinite Quivers
• Indecomposable representations of quivers on infinite-dimensional Hilbert spaces
• Decompositions of Infinite-Dimensional A _∞,∞ Quiver Representations
• Representation Theory of an Infinite Quiver
• The representations of quivers of type A_n. A fast approach