Picard groups of quasi-Frobenius algebras and a question on Frobenius strongly graded algebras
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866911956196130816 |
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| author | Dascalescu, Sorin Nastasescu, Constantin Nastasescu, Laura |
| author_facet | Dascalescu, Sorin Nastasescu, Constantin Nastasescu, Laura |
| contents | Our initial aim was to answer the question: does the Frobenius (symmetric) property transfers from a strongly graded algebra to its homogeneous component of trivial degree? Related to it, we investigate invertible bimodules and the Picard group of a finite dimensional quasi-Frobenius algebra $R$. We compute the Picard group, the automorphism group and the group of outer automorphisms of a $9$-dimensional quasi-Frobenius algebra which is not Frobenius, constructed by Nakayama. Using these results and a semitrivial extension construction, we give an example of a symmetric strongly graded algebra whose trivial homogeneous component is not even Frobenius. We investigate associativity of isomorphisms $R^*\ot_RR^*\simeq R$ for quasi-Frobenius algebras $R$, and we determine the order of the class of the invertible bimodule $H^*$ in the Picard group of a finite dimensional Hopf algebra $H$. As an application, we construct new examples of symmetric algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_10169 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Picard groups of quasi-Frobenius algebras and a question on Frobenius strongly graded algebras Dascalescu, Sorin Nastasescu, Constantin Nastasescu, Laura Rings and Algebras 16D50, 16D20, 16L60, 16S99, 16T05, 16W50 Our initial aim was to answer the question: does the Frobenius (symmetric) property transfers from a strongly graded algebra to its homogeneous component of trivial degree? Related to it, we investigate invertible bimodules and the Picard group of a finite dimensional quasi-Frobenius algebra $R$. We compute the Picard group, the automorphism group and the group of outer automorphisms of a $9$-dimensional quasi-Frobenius algebra which is not Frobenius, constructed by Nakayama. Using these results and a semitrivial extension construction, we give an example of a symmetric strongly graded algebra whose trivial homogeneous component is not even Frobenius. We investigate associativity of isomorphisms $R^*\ot_RR^*\simeq R$ for quasi-Frobenius algebras $R$, and we determine the order of the class of the invertible bimodule $H^*$ in the Picard group of a finite dimensional Hopf algebra $H$. As an application, we construct new examples of symmetric algebras. |
| title | Picard groups of quasi-Frobenius algebras and a question on Frobenius strongly graded algebras |
| topic | Rings and Algebras 16D50, 16D20, 16L60, 16S99, 16T05, 16W50 |
| url | https://arxiv.org/abs/2309.10169 |