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arXiv · 1612.01585

Preprojective algebras of tree-type quivers

Van C. Nguyen·Gordana Todorov·Shijie Zhu
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Abstract

Let $Q$ be a tree-type quiver, $\mathbf{k} Q$ its path algebra, and $\lambda$ a nonzero element in the field $\mathbf{k}$. We construct irreducible morphisms in the Auslander-Reiten quiver of the transjective component of the bounded derived category of $\mathbf{k} Q$ that satisfy what we call the $\lambda$-relations. When $\lambda=1$, the relations are known as mesh relations. When $\lambda=-1$, they are known as commutativity relations. Using this technique together with the results given by Baer-Geigle-Lenzing, Crawley-Boevey, Ringel, and others, we show that for any tree-type quiver, several descriptions of its preprojective algebra are equivalent.

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Van C. Nguyen, Gordana Todorov, Shijie Zhu. 2016-12-05. Preprojective algebras of tree-type quivers. https://arxiv.org/abs/1612.01585

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