Mathematics Colloquia and Seminars

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We will discuss two versions of Gabriel's theorem for infinite quivers. More precisely, we show that (1) the category of all (possibly infinite dimensional) representations of a quiver $\Omega$ is of unique type (each dimension vector has at most one associated indecomposable) and infinite Krull-Schmidt (every representation is a direct sum of indecomposables) if and only if $\Omega$ is eventually outward and of generalized ADE Dynkin type ($A_n$, $D_n$, $E_6$, $E_7$, $E_8$, $A_\infty$, $A_{\infty, \infty}$, or $D_\infty$) and (2) restricting to the category of locally finite-dimensional representations of $\Omega$ allows us to relax the eventually outward condition, i.e. the restricted category is of unique type if and only if $\Omega$ is of generalized ADE Dynkin type. 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 $1$). This is joint work with Stephen Sawin.