Infinitesimal $\mathcal{R}$-matrices for some families of Hopf algebras
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912753714724864 |
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| author | Bottegoni, Lucrezia Renda, Fabio Sciandra, Andrea |
| author_facet | Bottegoni, Lucrezia Renda, Fabio Sciandra, Andrea |
| contents | Given a bialgebra $H$ such that the associated trivial topological bialgebra $H[[\hbar]]$ admits a quasitriangular structure $\tilde{\mathcal{R}}=\mathcal{R}(1\otimes 1+\hbarχ+\mathcal{O}(\hbar^2))$, one gets a distinguished element $χ\in H \otimes H$ which is an infinitesimal $\mathcal{R}$-matrix, according to the definition given in [1]. In this paper we classify infinitesimal $\mathcal{R}$-matrices for some families of well-known Hopf algebras, among which are the generalized Kac-Paljutkin Hopf algebras $H_{2n^2}$, the Radford Hopf algebras $H_{(r,n,q)}$, and the Hopf algebras $E(n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_02350 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Infinitesimal $\mathcal{R}$-matrices for some families of Hopf algebras Bottegoni, Lucrezia Renda, Fabio Sciandra, Andrea Quantum Algebra Rings and Algebras Representation Theory Primary 16T05, Secondary 16E40 Given a bialgebra $H$ such that the associated trivial topological bialgebra $H[[\hbar]]$ admits a quasitriangular structure $\tilde{\mathcal{R}}=\mathcal{R}(1\otimes 1+\hbarχ+\mathcal{O}(\hbar^2))$, one gets a distinguished element $χ\in H \otimes H$ which is an infinitesimal $\mathcal{R}$-matrix, according to the definition given in [1]. In this paper we classify infinitesimal $\mathcal{R}$-matrices for some families of well-known Hopf algebras, among which are the generalized Kac-Paljutkin Hopf algebras $H_{2n^2}$, the Radford Hopf algebras $H_{(r,n,q)}$, and the Hopf algebras $E(n)$. |
| title | Infinitesimal $\mathcal{R}$-matrices for some families of Hopf algebras |
| topic | Quantum Algebra Rings and Algebras Representation Theory Primary 16T05, Secondary 16E40 |
| url | https://arxiv.org/abs/2412.02350 |