Picard groups of quasi-Frobenius algebras and a question on Frobenius strongly graded algebras

Autores: Sorin Dascalescu, Constantin Nastasescu , Laura Nastasescu
Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 70, Nº 1, 2026, págs. 3-25
Idioma: inglés
DOI: 10.5565/publmat7012601
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Resumen
- 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.