A linear algebra approach to graded Frobenius algebras

TL;DR

This paper introduces a linear algebra method using paratrophic matrices to characterize and analyze the graded Frobenius and symmetric properties of finite-dimensional group-graded algebras, with applications to quantum polynomial algebras.

Contribution

It develops a novel linear algebra framework for studying graded Frobenius algebras, providing new tools for their characterization and analysis.

Findings

Characterization of $\sigma$-graded Frobenius property using paratrophic matrices.

Criteria for invertibility of paratrophic matrices related to algebra properties.

Identification of Frobenius and symmetric properties in Koszul duals of quantum polynomial algebras.

Abstract

If $A$ is a finite-dimensional algebra graded by a group $G$, and $\sigma \in G$, we define a variant of paratrophic matrix associated with $A$ and $\sigma$, and we use it to characterize the $\sigma$-graded Frobenius property for $A$. We discuss the invertibility of such paratrophic matrices, and then use them to check whether certain graded algebras are $\sigma$-graded Frobenius or (graded) symmetric. As an application, we uncover (graded) Frobenius and symmetric properties of Koszul duals of quantum polynomial algebras. We derive a structure result for $\sigma$-graded Frobenius algebras by only using linear algebra methods.