Bi-Frobeniusu Algebras as a Generalization of Finite-dimensional Hopf Algebras

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Bi-Frobenius algebras (or bF algebras) were introduced by the author and Takeuchi in 2000. These are both Frobenius algebras and Frobenius coalgebras and satisfy some compatibility conditions. The concept generalizes finite, dimensional Hopf algebras. In this research project we studied substructures, quotient structures and morphisms of bi-Frobenius algebras. We also gave that -under certain assumptions-the irreducible characters are orthogonal.We next introduced and studied a notion of a group-like algebra, which is a finite dimensional algebra A over a field k with a distinguished k-basis B with some conditions. The concept generalizes Bose-Mesner algebras of non-commutative association schemes. Group-like algebras become bF algebras in a natural way.We determined the structure of group-like algebras of dimension 2 or 3, and showed that group-like algebras of dimension 4 or 5 are commutative algebras. We also considered a subset N ⊂ B such that the k-span kN forms a group-like algebra. We determined the structure of the ring extension kN ⊂ A in the case of |N| = |B| -1.