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Algebra/Topology Seminar – Heather Werth (University of Alabama)
September 29, 2021 @ 11:00 am - 12:00 pm
Title: Computation of extension spaces in $kQ$-mod, for $kQ$ the path algebra of a quiver $Q$ of type $\tilde A(n-1,1)$, using planar curves.
Abstract: The representation theory of quivers is important to the representation theory of associative algebras in general. If $Q$ is a quiver of affine type $\tilde A(n-1,1)$ and $k$ a fixed algebraically closed field, one defines the path algebra $kQ$ and the module category $kQ$-mod. The self-extension space of a given object $M$ in $kQ$-mod is generally difficult to compute and understand, but we show that its dimension can be deduced from the properties of a particular plane curve. Specifically, it is true that for a large class of modules, $M$ corresponds to a curve $\gamma(M)$ whose number of self-intersections is equal to the dimension of $M$’s self extension space.
We explain this correspondence and its relationships to the broad fields of representation theory, combinatorics, and homological algebra. We also discuss extensions of this result which allow for computation of extension spaces in the cluster category of $kQ$-mod.