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arXiv · 2309.10169

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

Sorin Dascalescu·Constantin Nastasescu·Laura Nastasescu
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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.

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Sorin Dascalescu, Constantin Nastasescu, Laura Nastasescu. 2023-09-18. Picard groups of quasi-Frobenius algebras and a question on Frobenius strongly graded algebras. https://arxiv.org/abs/2309.10169

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