Elementary modules of the generalized Kronecker quivers K(n)
Abstract
Let k be an algebraically closed field. The generalized or n-Kronecker quiver K(n) is the quiver with two vertices, called a source and a sink, and n arrows from source to sink. We use moddcKn to denote the category of the finite-dimensional modules of the path algebra cKn=kK(n). There exist a function called dimension vector dimu: modd cKn→ ℤ2; M ↦ (dimk M1, dimk M2 ), and two functors σ: ℤ2 → ℤ2; (x,y)↦ (nx−y,x), and δ: ℤ2→ ℤ2;(x,y)↦ (y,x). Then the set F={(x,y)| 2/nx≤ y≤ x} is the fundamental domain of the dimension vectors of regular modules under the action of δ and σ in moddcKn. We call a regular module M∈ moddcKn \textit{elementary} if there does not exist a short exact sequence (0)→ L→ M→ N→ (0) with L,N being non-zero regular. In this note, we focus on the set F, and when x<n, we establish a one-to-one correspondence between elementary modules and spaces of fixed rank.
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BibTeX
@misc{liu2026elementary,
title = {{Elementary modules of the generalized Kronecker quivers K(n)}},
author = {Jie Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14238},
eprint = {2609.14238},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14238},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Elementary modules of the generalized Kronecker quivers K(n)
AU - Jie Liu
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14238
UR - https://arxiv.org/abs/2609.14238
AB - Let k be an algebraically closed field. The generalized or n-Kronecker quiver K(n) is the quiver with two vertices, called a source and a sink, and n arrows from source to sink. We use moddcK_n to denote the category of the finite-dimensional modules of the path algebra cK_n=kK(n). There exist a function called dimension vector dimu: modd cK_n→ ℤ^2; M ↦ (dim_k M_1, dim_k M_2 ), and two functors σ: ℤ^2 → ℤ^2; (x,y)↦ (nx−y,x), and δ: ℤ^2→ ℤ^2;(x,y)↦ (y,x). Then the set F={(x,y)| 2/nx≤ y≤ x} is the fundamental domain of the dimension vectors of regular modules under the action of δ and σ in moddcK_n. We call a regular module M∈ moddcK_n \textit{elementary} if there does not exist a short exact sequence (0)→ L→ M→ N→ (0) with L,N being non-zero regular. In this note, we focus on the set F, and when x<n, we establish a one-to-one correspondence between elementary modules and spaces of fixed rank.
ER -