PreprintCC BY 4.0arXiv
Picture groups and green sequences from the perspective of Ringel--Hall algebras
Abstract
Let k be an algebraically closed field and let Λ be a finite-dimensional associative k-algebra. We apply Joyce and Brideland's notion of Ringel--Hall algebra to prove results about picture spaces and picture groups. For instance, we construct a faithful group functor for Λ, i.e. a faithful functor from the τ-cluster morphism category of Λ into a groupoid. Consequently, by results of Hanson--Igusa, the picture space of Λ is locally CAT(0) provided that the τ-cluster morphism category of Λ admits compatibility of last factors. We also show that green sequences are in bijection with certain positive expressions in the picture group, which generalizes a result of Igusa--Todorov beyond hereditary k-algebras.
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BibTeX
@misc{borve2026picture,
title = {{Picture groups and green sequences from the perspective of Ringel--Hall algebras}},
author = {Erlend D. Børve},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29960},
eprint = {2608.29960},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29960},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Picture groups and green sequences from the perspective of Ringel--Hall algebras
AU - Erlend D. Børve
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29960
UR - https://arxiv.org/abs/2608.29960
AB - Let k be an algebraically closed field and let Λ be a finite-dimensional associative k-algebra. We apply Joyce and Brideland's notion of Ringel--Hall algebra to prove results about picture spaces and picture groups. For instance, we construct a faithful group functor for Λ, i.e. a faithful functor from the τ-cluster morphism category of Λ into a groupoid. Consequently, by results of Hanson--Igusa, the picture space of Λ is locally CAT(0) provided that the τ-cluster morphism category of Λ admits compatibility of last factors. We also show that green sequences are in bijection with certain positive expressions in the picture group, which generalizes a result of Igusa--Todorov beyond hereditary k-algebras.
ER -