Quantum group deformations and quantum $ R $-(co)matrices vs. Quantum Duality Principle
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915480581701632 |
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| author | García, Gastón Andrés Gavarini, Fabio |
| author_facet | García, Gastón Andrés Gavarini, Fabio |
| contents | In this paper we describe the effect on quantum groups -- namely, both QUEA's and QFSHA's -- of deformations by twist and by 2-cocycles, showing how such deformations affect the semiclassical limit. As a second, more important task, we discuss how these deformation procedures can be "stretched" to a new extent, via a formal variation of the original recipes, using "polar twists" and "polar 2-cocycles". These recipes seemingly should make no sense at all, yet we prove that they actually work, thus providing well-defined, more general deformation procedures. Later on, we explain the underlying reason that motivates such a result in light of the "Quantum Duality Principle", through which every "polar twist/2-cocycle" for a given quantum group can be seen as a standard twist/2-cocycle for another quantum group, associated to the original one via the appropriate Drinfeld functor. As a third task, we consider standard constructions involving $R$-(co)matrices in the general theory of Hopf algebras. First we adapt them to quantum groups, then we show that they extend to the case of "polar $R$-(co)matrices", and finally we discuss how these constructions interact with the Quantum Duality Principle. As a byproduct, this yields new special symmetries (isomorphisms) for the underlying pair of dual Poisson (formal) groups that one gets by specialization. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_15096 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantum group deformations and quantum $ R $-(co)matrices vs. Quantum Duality Principle García, Gastón Andrés Gavarini, Fabio Quantum Algebra Rings and Algebras 17B37, 17B62 In this paper we describe the effect on quantum groups -- namely, both QUEA's and QFSHA's -- of deformations by twist and by 2-cocycles, showing how such deformations affect the semiclassical limit. As a second, more important task, we discuss how these deformation procedures can be "stretched" to a new extent, via a formal variation of the original recipes, using "polar twists" and "polar 2-cocycles". These recipes seemingly should make no sense at all, yet we prove that they actually work, thus providing well-defined, more general deformation procedures. Later on, we explain the underlying reason that motivates such a result in light of the "Quantum Duality Principle", through which every "polar twist/2-cocycle" for a given quantum group can be seen as a standard twist/2-cocycle for another quantum group, associated to the original one via the appropriate Drinfeld functor. As a third task, we consider standard constructions involving $R$-(co)matrices in the general theory of Hopf algebras. First we adapt them to quantum groups, then we show that they extend to the case of "polar $R$-(co)matrices", and finally we discuss how these constructions interact with the Quantum Duality Principle. As a byproduct, this yields new special symmetries (isomorphisms) for the underlying pair of dual Poisson (formal) groups that one gets by specialization. |
| title | Quantum group deformations and quantum $ R $-(co)matrices vs. Quantum Duality Principle |
| topic | Quantum Algebra Rings and Algebras 17B37, 17B62 |
| url | https://arxiv.org/abs/2403.15096 |