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On Hopf algebras with positive bases

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J.H. Lu, M. Yan, Y.C. Zhu

Quantum Algebra · Submitted 13 Nov 1999

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We show that if a finite dimensional Hopf algebra over ${\bf C}$ has a basis such that all the structure constants are non-negative, then the Hopf algebra must be given by a finite group $G$ and a factorization $G=G_+G_-$ into two subgroups.

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arXiv IDmath/9911091 (version 1)
Pith IDWXVNFUD2, Pith's permanent record of this paper and its review
CitationJ.H. Lu, M. Yan, Y.C. Zhu. "On Hopf algebras with positive bases." arXiv:math/9911091 (1999). https://pith.science/paper/WXVNFUD2
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,003 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.09 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.27 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We show that if a finite dimensional Hopf algebra over ${\bf C}$ has a basis such that all the structure constants are non-negative, then the Hopf algebra must be given by a finite group $G$ and a factorization $G=G_+G_-$ into two subgroups. We also show that Hopf algebras in the category of finite sets with correspondences as morphisms are classified in the similar way. Our results can be used to explain some results in Hopf algebras from set-theoretical viewpoint.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We show that if a finite dimensional Hopf algebra over ${\bf C}$ has a basis such that all the structure constants are non-negative, then the Hopf algebra must be given by a finite group $G$ and a factorization $G=G_+G_-$ into two subgroups. result 0.99 0.96 0.86 asserted 1.00
s1 We also show that Hopf algebras in the category of finite sets with correspondences as morphisms are classified in the similar way. result 0.97 0.93 0.09 asserted 1.00
s2 Our results can be used to explain some results in Hopf algebras from set-theoretical viewpoint. interpretation 1.00 0.65 0.05 hedged possibility 0.72

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We show that if a finite dimensional Hopf algebra over ${\bf C}$ has a basis such that all the structure constants are non-negative, then the Hopf algebra must be given by a finite group $G$ and a factorization $G=G_+G_-$ into two subgroups. We also show that Hopf algebras in the category of finite sets with correspondences as morphisms are classified in the similar way. Our results can be used to explain some results in Hopf algebras from set-theoretical viewpoint.

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