Abstract
We show that any extension of an Abelian group corresponds to a solution of the parametric Yang–Baxter equation. This statement is a generalization of the well-known construction of a braided set in terms of group structure to the case of group extensions. We also show that this construction in the case of a semidirect product is a specialization of a more general construction using principal bundles and that the case of vector bundles considered earlier is an infinitesimal version of the case of a solution coming from the principal bundle structure.
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Notes
It is also called a shelf by topologists (see, e.g., [13]).
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Acknowledgments
The authors express their gratitude to participants of the work group “Algebraic structures in discrete integrable systems” of the regional mathematical center of Demidov Yaroslavl State University and especially to S. Konstantinou-Rizos and S. Igoninn for the fruitful discussions.
Funding
This research was supported by a grant from the Russian Science Foundation (Project No. 20-71-10110).
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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2021, Vol. 207, pp. 310-318 https://doi.org/10.4213/tmf10022.
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Preobrazhenskaya, M.M., Talalaev, D.V. Group extensions, fiber bundles, and a parametric Yang–Baxter equation. Theor Math Phys 207, 670–677 (2021). https://doi.org/10.1134/S004057792105010X
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DOI: https://doi.org/10.1134/S004057792105010X