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Representations of Regular Trees and Invariants of AR-Components for Generalized Kronecker Quivers

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Abstract

We investigate the generalized Kronecker algebra 𝒦 r = kΓ r with r ≥ 3 arrows. Given a regular component 𝒞 of the Auslander-Reiten quiver of 𝒦 r , we show that the quasi-rank rk(𝒞) ∈ ℤ≤1 can be described almost exactly as the distance 𝒲(𝒞) ∈ ℕ0 between two non-intersecting cones in 𝒞, given by modules with the equal images and the equal kernels property; more precisley, we show that the two numbers are linked by the inequality

$$-\mathcal{W}(\mathcal{C}) \leq \text{rk}(\mathcal{C}) \leq - \mathcal{W}(\mathcal{C}) + 3.$$

Utilizing covering theory, we construct for each n ∈ ℕ0 a bijection φ n between the field k and {𝒞∣𝒞 regular component, 𝒲(𝒞) = n}. As a consequence, we get new results about the number of regular components of a fixed quasi-rank.

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Acknowledgements

The results of this article are part of my doctoral thesis, which I am currently writing at the University of Kiel. I would like to thank my advisor Rolf Farnsteiner for his continuous support and helpful comments. I also would like to thank the whole research team for the very pleasant working atmosphere and the encouragement throughout my studies. In particular, I thank Christian Drenkhahn for proofreading.

Furthermore, I thank Claus Michael Ringel for fruitful discussions during my visits in Bielefeld. I also would like to thank the anonymous referee for the detailed comments that helped to improve the exposition of this article.

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  1. Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, 24118, Kiel, Germany

    Daniel Bissinger

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  1. Daniel Bissinger
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Correspondence to Daniel Bissinger.

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Presented by Henning Krause.

Partly supported by the D.F.G. priority program SPP 1388 “Darstellungstheorie”

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Bissinger, D. Representations of Regular Trees and Invariants of AR-Components for Generalized Kronecker Quivers. Algebr Represent Theor 21, 331–358 (2018). https://doi.org/10.1007/s10468-017-9716-x

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