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Quivers and the Euclidean algebra (Extended abstract)

Abstract : We show that the category of representations of the Euclidean group $E(2)$ is equivalent to the category of representations of the preprojective algebra of the quiver of type $A_{\infty}$. Furthermore, we consider the moduli space of $E(2)$-modules along with a set of generators. We show that these moduli spaces are quiver varieties of the type considered by Nakajima. These identifications allow us to draw on known results about preprojective algebras and quiver varieties to prove various statements about representations of $E(2)$. In particular, we show that $E(2)$ has wild representation type but that if we impose certain combinatorial restrictions on the weight decompositions of a representation, we obtain only a finite number of indecomposable representations.
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Alistair Savage. Quivers and the Euclidean algebra (Extended abstract). 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), 2008, Viña del Mar, Chile. pp.417-424. ⟨hal-01185136⟩

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