2023/09/18 by Dascalescu, Sorin, Nastasescu, Constantin, Nastasescu, Laura
#16D20 #16D50 #16L60 #16S99 #16T05 #16W50 #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2309.10169
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^*\otRR^*≃ 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.