Graded Frobenius algebras from tensor algebras of bimodules
Sorin Dascalescu, Constantin Nastasescu, Laura Nastasescu
TL;DR
The paper constructs and analyzes algebras $A(R,M,\varphi)$ obtained from tensor algebras of bimodules by quotienting with an associative bimodule map $\varphi$, yielding a natural $\mathbb{Z}_n$-grading. It develops a framework for graded Frobenius theory, classifies associative morphisms, and derives precise criteria for when these algebras are Frobenius or symmetric, including the key special case $M=R^*$ and various Morita-transfer techniques. The authors prove that strong-gradability and Frobenius properties arise under invertibility and isomorphism conditions on tensor powers of $M$, and they provide detailed theorems (A–F) that connect Nakayama automorphisms, projectivity, and Picard-group data to the resulting graded Frobenius structure. The results yield a broad class of symmetric and Frobenius algebras with explicit descriptions of Frobenius forms and Nakayama automorphisms, and they are illustrated by several examples, including quantum planes and Hopf-algebra contexts. This extends known trivial/semitrivial extensions and offers a scalable approach to constructing (graded) Frobenius algebras from bimodule tensor algebras with applications to noncommutative geometry and representation theory.
Abstract
We consider certain quotient algebras of tensor algebras of bimodules $M$ over a finite-dimensional algebra $R$, and we investigate Frobenius type properties of such algebras. Our main interest is in the case where $M=R^*$, the linear dual of $R$. We obtain a large class of Frobenius or symmetric algebras, which are also equipped with a finite grading.