Frobenius Structural Matrix Algebras

TL;DR

This paper characterizes when structural matrix algebras are Frobenius, showing they are Frobenius if and only if they are block-diagonal with full matrix blocks, based on incidence coalgebra properties.

Contribution

It provides a precise criterion for Frobenius property in structural matrix algebras using incidence coalgebra theory.

Findings

Structural matrix algebra is Frobenius iff it has a block-diagonal form with full matrix blocks.

The incidence coalgebra of a locally finite preordered set is right co-Frobenius under certain conditions.

The characterization links algebraic Frobenius property to combinatorial block structures.

Abstract

We discuss when the incidence coalgebra of a locally finite preordered set is right co-Frobenius. As a consequence, we obtain that a structural matrix algebra over a field $k$ is Frobenius if and only if it consists, up to a permutation of rows and columns, of diagonal blocks which are full matrix algebras over $k$.