Picard groups of quasi-Frobenius algebras and a question on Frobenius strongly graded algebras
Our initial aim was to answer the question: does the Frobenius (symmetric) property transfer 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 co...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2026 |
| País: | España |
| Institución: | Universitat Autònoma de Barcelona |
| Repositorio: | Dipòsit Digital de Documents de la UAB |
| Idioma: | inglés |
| OAI Identifier: | oai:ddd.uab.cat:324076 |
| Acceso en línea: | https://ddd.uab.cat/record/324076 https://dx.doi.org/urn:doi:10.5565/PUBLMAT7012601 |
| Access Level: | acceso abierto |
| Palabra clave: | Quasi-frobenius algebra Frobenius algebra Symmetric algebra Invertible bimodule Picard group Strongly graded algebra Hopf algebra Nakayama automorphism |
| Sumario: | Our initial aim was to answer the question: does the Frobenius (symmetric) property transfer 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∗⊗RR∗ ≃ 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. |
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