On a class of quasi-Frobenius algebras (Q2078411)

In this paper, the authors construct finite dimensional coalgebras by starting with two matrix coalgebras \(\mathcal{E}\) and \(\mathcal{F}\), and then by adjoining two families of elements which in a matrix-formal way behave like some kind of skew-primitives with respect to \(\mathcal{E}\) and \(\mathcal{F}\). Then it is shown that the obtained coalgebras are quasi-co-Frobenius. Further, it is shown that the obtained colagebras, which are quasi-co-Frobenius, become co-Frobenius only when \(\mathcal{E}\) and \(\mathcal{F}\) have the same size. By taking the dual algebras, examples of quasi-Frobenius algebras which are not Frobenius are obtained.