Algebraic properties of quantum quasigroups
| Autor: | Alex W. Nowak, Jonathan D. H. Smith, Bokhee Im |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Pure mathematics
Algebra and Number Theory Triality Mathematics::General Mathematics Direct sum Distributivity 010102 general mathematics Hopf algebra 01 natural sciences Mathematics::Group Theory Transfer (group theory) Quantization (physics) Distributive property Mathematics::Quantum Algebra 0103 physical sciences 010307 mathematical physics 0101 mathematics Algebraic number Computer Science::Cryptography and Security Mathematics |
| Zdroj: | Journal of Pure and Applied Algebra. 225:106539 |
| ISSN: | 0022-4049 |
| DOI: | 10.1016/j.jpaa.2020.106539 |
| Popis: | Quantum quasigroups provide a self-dual framework for the unification of quasigroups and Hopf algebras. This paper furthers the transfer program, investigating extensions to quantum quasigroups of various algebraic features of quasigroups and Hopf algebras. Part of the difficulty of the transfer program is the fact that there is no standard model-theoretic procedure for accommodating the coalgebraic aspects of quantum quasigroups. The linear quantum quasigroups, which live in categories of modules under the direct sum, are a notable exception. They form one of the central themes of the paper. From the theory of Hopf algebras, we transfer the study of grouplike and setlike elements, which form separate concepts in quantum quasigroups. From quasigroups, we transfer the study of conjugate quasigroups, which reflect the triality symmetry of the language of quasigroups. In particular, we construct conjugates of cocommutative Hopf algebras. Semisymmetry, Mendelsohn, and distributivity properties are formulated for quantum quasigroups. We classify distributive linear quantum quasigroups that furnish solutions to the quantum Yang-Baxter equation. The transfer of semisymmetry is designed to prepare for a quantization of web geometry. |
| Databáze: | OpenAIRE |
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