Weak Hopf quasigroups and matched pairs of quasigroupoids
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| Format: | Preprint |
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2024
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| _version_ | 1866913264624992256 |
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| author | Rodríguez, Ramón González |
| author_facet | Rodríguez, Ramón González |
| contents | In this paper we introduce the notion of exact factorization of a quasigroupoid and the notion of matched pair of quasigroupoids with common base. We prove that if $({\sf A}, {\sf H})$ is a matched pair of quasigroupoids it is posible to construct a new quasigroupoid ${\sf A}\bowtie {\sf H}$ called the double cross product of ${\sf A}$ and ${\sf H}$. Also, we show that, if a quasigroupoid ${\sf B}$ admits an exact factorization, there exists a matched pair of quasigroupoids $({\sf A}, {\sf H})$ and an isomorphism of quasigroupoids between ${\sf A}\bowtie {\sf H}$ and ${\sf B}$. Finally, if ${\mathbb K}$ is a field, we show that every matched pair of quasigroupoids $({\sf A}, {\sf H})$ induce, thanks to the quasigroupoid magma construction, a pair $({\mathbb K}[{\sf A}], {\mathbb K}[{\sf H}])$ of weak Hopf quasigroups and a double crossed product weak Hopf quasigroup ${\mathbb K}[{\sf A}]\bowtie{\mathbb K}[{\sf H}]$ isomorphic to ${\mathbb K}[{\sf A}\bowtie {\sf H}]$ as weak Hopf quasigroups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_09231 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weak Hopf quasigroups and matched pairs of quasigroupoids Rodríguez, Ramón González Rings and Algebras 20N05, 16T05, 17A01 In this paper we introduce the notion of exact factorization of a quasigroupoid and the notion of matched pair of quasigroupoids with common base. We prove that if $({\sf A}, {\sf H})$ is a matched pair of quasigroupoids it is posible to construct a new quasigroupoid ${\sf A}\bowtie {\sf H}$ called the double cross product of ${\sf A}$ and ${\sf H}$. Also, we show that, if a quasigroupoid ${\sf B}$ admits an exact factorization, there exists a matched pair of quasigroupoids $({\sf A}, {\sf H})$ and an isomorphism of quasigroupoids between ${\sf A}\bowtie {\sf H}$ and ${\sf B}$. Finally, if ${\mathbb K}$ is a field, we show that every matched pair of quasigroupoids $({\sf A}, {\sf H})$ induce, thanks to the quasigroupoid magma construction, a pair $({\mathbb K}[{\sf A}], {\mathbb K}[{\sf H}])$ of weak Hopf quasigroups and a double crossed product weak Hopf quasigroup ${\mathbb K}[{\sf A}]\bowtie{\mathbb K}[{\sf H}]$ isomorphic to ${\mathbb K}[{\sf A}\bowtie {\sf H}]$ as weak Hopf quasigroups. |
| title | Weak Hopf quasigroups and matched pairs of quasigroupoids |
| topic | Rings and Algebras 20N05, 16T05, 17A01 |
| url | https://arxiv.org/abs/2403.09231 |