Triangle Relation Planar Algebras: Cohomology, Classification, and New Families
Abstract
We establish a finite system of consistency equations for triangle relation planar algebras. Within the Yang--Baxter relation class, we characterize triangle relations by H2(K;ℂ)=0, where K is the fusion complex recording the support of the fusion coefficients. Together with a reconstruction theorem, this gives an algebraic classification procedure. We classify bi-commutative triangle relation planar algebras with dim P2,±≤5. In ranks two, three, and four, the classification yields respectively $2$, $7$, and $19$ realizable fusion-support pair strata, up to relabelling and shading reversal. These results recover the low-rank examples of Bisch, Jones, Liu, and Lu, and extend the rank-four classification beyond exchange relations. In rank five, we obtain two one-parameter families with different duality types whose generic members do not arise from representations of groups or quantum groups.
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BibTeX
@misc{li2026triangle,
title = {{Triangle Relation Planar Algebras: Cohomology, Classification, and New Families}},
author = {Jiaqi Li and Zhengwei Liu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02714},
eprint = {2611.02714},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02714},
}
RIS
TY - UNPB
TI - Triangle Relation Planar Algebras: Cohomology, Classification, and New Families
AU - Jiaqi Li
AU - Zhengwei Liu
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.02714
UR - https://arxiv.org/abs/2611.02714
AB - We establish a finite system of consistency equations for triangle relation planar algebras. Within the Yang--Baxter relation class, we characterize triangle relations by H^2(K;ℂ)=0, where K is the fusion complex recording the support of the fusion coefficients. Together with a reconstruction theorem, this gives an algebraic classification procedure. We classify bi-commutative triangle relation planar algebras with dim P_2,±≤5. In ranks two, three, and four, the classification yields respectively $2$, $7$, and $19$ realizable fusion-support pair strata, up to relabelling and shading reversal. These results recover the low-rank examples of Bisch, Jones, Liu, and Lu, and extend the rank-four classification beyond exchange relations. In rank five, we obtain two one-parameter families with different duality types whose generic members do not arise from representations of groups or quantum groups.
ER -