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PreprintCC BY-NC-ND 4.0arXiv

Representations of Bound Quivers over 𝔽1 and Hall Algebras

  • Changjian Fu
  • Xuelian Xiao
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Abstract

Representations of quivers over the virtual field 𝔽1 can be naturally extended to bound quivers with monomial and commutativity relations. In this paper, we introduce and study the Hall algebra ℋ𝔽1(Q,I) for a bound quiver (Q,I) equipped with such relations. As in the quiver setting, ℋ𝔽1(Q,I) is a graded, connected, and cocommutative bialgebra, and the free subgroup generated by all indecomposable representations forms a Lie subalgebra. We study ℋ𝔽1(Q,I) in detail for two special classes of bound quivers: Nakayama bound quivers and gentle one-cycle bound quivers. In both cases, we prove that for any triple $L, M, N$ of 𝔽1-representations over (Q,I), the Hall number gN,LM coincides with the evaluation of the corresponding Hall polynomial Ο•N,LM(t) at t=1. Consequently, there exists a canonical isomorphism between ℋ𝔽1(Q,I) and the degenerate Hall algebra β„‹1(Q,I), justifying Tits's 𝔽1-philosophy in these settings.

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Cite

BibTeXRIS
BibTeX
@misc{fu2026representations,
  title = {{Representations of Bound Quivers over $\mathbbF_1$ and Hall Algebras}},
  author = {Changjian Fu and Xuelian Xiao},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.02904},
  eprint = {2611.02904},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.02904},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Representations of Bound Quivers over 𝔽_1 and Hall Algebras
AU  - Changjian Fu
AU  - Xuelian Xiao
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2611.02904
UR  - https://arxiv.org/abs/2611.02904
AB  - Representations of quivers over the virtual field 𝔽_1 can be naturally extended to bound quivers with monomial and commutativity relations. In this paper, we introduce and study the Hall algebra β„‹_𝔽_1(Q,I) for a bound quiver (Q,I) equipped with such relations. As in the quiver setting, β„‹_𝔽_1(Q,I) is a graded, connected, and cocommutative bialgebra, and the free subgroup generated by all indecomposable representations forms a Lie subalgebra. We study β„‹_𝔽_1(Q,I) in detail for two special classes of bound quivers: Nakayama bound quivers and gentle one-cycle bound quivers. In both cases, we prove that for any triple $L, M, N$ of 𝔽_1-representations over (Q,I), the Hall number g_N,L^M coincides with the evaluation of the corresponding Hall polynomial Ο•_N,L^M(t) at t=1. Consequently, there exists a canonical isomorphism between β„‹_𝔽_1(Q,I) and the degenerate Hall algebra β„‹_1(Q,I), justifying Tits's 𝔽_1-philosophy in these settings.
ER  -