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

TL;DR

This paper explores the transfer of Frobenius properties in strongly graded algebras, computes Picard groups for quasi-Frobenius algebras, and constructs new symmetric algebra examples, addressing a key open question.

Contribution

It provides the first detailed computation of Picard groups for certain quasi-Frobenius algebras and constructs examples showing Frobenius properties do not always transfer.

Findings

Computed Picard, automorphism, and outer automorphism groups for a 9-dimensional quasi-Frobenius algebra.

Constructed a symmetric strongly graded algebra with a non-Frobenius trivial component.

Determined the order of the invertible bimodule class in the Picard group of a finite dimensional Hopf algebra.

Abstract

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.