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Cotangent and tangent modules on quantum orbits

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P.Akueson, D. Gurevich (Universite de Valenciennes)

Quantum Algebra · Submitted 11 May 2000

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

The authors' main claim This implies non-flatness of any covariant differential calculus on the quantum sphere making use of the Leibniz rule.

how it is stated · asserted 0.99 read the record →

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arXiv IDmath/0005109 (version 1)
Pith ID3ZAYE547, Pith's permanent record of this paper and its review
CitationD. Gurevich (Universite de Valenciennes), P.Akueson. "Cotangent and tangent modules on quantum orbits." arXiv:math/0005109 (2000). https://pith.science/paper/3ZAYE547
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,642 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.79 · human participants 0.01 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.24 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

Let $k(S^2_q)$ be the "coordinate ring" of a quantum sphere. We introduce the cotangent module on the quantum sphere as a one-sided $k(S^2_q)$-module and show that there is no Yang-Baxter type operator converting it into a $k(S^2_q)$-bimodule which would be a flatly deformed object w.r.t. its classical counterpart. This implies non-flatness of any covariant differential calculus on the quantum sphere making use of the Leibniz rule. Also, we introduce the cotangent and tangent modules on generic quantum orbits and discuss some related problems of "braided geometry".

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 Let $k(S^2_q)$ be the "coordinate ring" of a quantum sphere. background 0.99 0.15 0.02 asserted 0.95
s1 We introduce the cotangent module on the quantum sphere as a one-sided $k(S^2_q)$-module and show that there is no Yang-Baxter type operator converting it into a $k(S^2_q)$-bimodule which would be a flatly deformed object w.r.t. its classical counterpart. contribution 0.69 0.95 0.68 asserted 1.00
s2 This implies non-flatness of any covariant differential calculus on the quantum sphere making use of the Leibniz rule. interpretation 0.98 0.80 0.54 asserted 0.99
s3 Also, we introduce the cotangent and tangent modules on generic quantum orbits and discuss some related problems of "braided geometry". contribution 1.00 0.94 0.07 asserted 0.97

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

Let $k(S^2_q)$ be the "coordinate ring" of a quantum sphere. We introduce the cotangent module on the quantum sphere as a one-sided $k(S^2_q)$-module and show that there is no Yang-Baxter type operator converting it into a $k(S^2_q)$-bimodule which would be a flatly deformed object w.r.t. its classical counterpart. This implies non-flatness of any covariant differential calculus on the quantum sphere making use of the Leibniz rule. Also, we introduce the cotangent and tangent modules on generic quantum orbits and discuss some related problems of "braided geometry".

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