Picard groups of quasi-Frobenius algebras and a question on Frobenius strongly graded algebras
Dascalescu, Sorin (University of Bucharest. Faculty of Mathematics and Computer Science)
Nastasescu, Constantin (Romanian Academy (Bucarest, Romania). Institute of Mathematics)
Nastasescu, Laura (Romanian Academy (Bucarest, Romania). Institute of Mathematics)

Data:	2026
Resum:	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.
Nota:	The first two authors were supported by a grant from UEFISCDI, project num-ber PN-III-P4-PCE-2021-0282, contract PCE 47/2022
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Llengua:	Anglès
Document:	Article ; recerca ; Versió publicada
Matèria:	Quasi-frobenius algebra ; Frobenius algebra ; Symmetric algebra ; Invertible bimodule ; Picard group ; Strongly graded algebra ; Hopf algebra ; Nakayama automorphism
Publicat a:	Publicacions matemàtiques, Vol. 70, Num. 1 (2026) , p. 3-25 (Articles) , ISSN 2014-4350

Adreça original: https://raco.cat/index.php/PublicacionsMatematiques/article/view/980000003194
DOI: 10.5565/PUBLMAT7012601

23 p, 1.1 MB

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