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

Bi-Frobenius代数（或bF代数）是作者和Takeuchi在2000年引入的。它们都是Frobenius代数和Frobenius余代数，并且满足一些相容条件。这个概念推广了有限维Hopf代数。本课题研究了双frobenius代数的子结构、商结构和态态。我们也给出了——在某些假设下——不可约的性质是正交的。接下来，我们引入并研究了类群代数的概念，它是域k上的有限维代数a，具有不同的k基B，并具有一定的条件。该概念推广了非交换关联格式的玻色-梅斯纳代数。类群代数自然地变成bF代数。我们确定了2维和3维类群代数的结构，并证明了4维和5维类群代数是交换代数。我们还考虑了一个子集N∧B，使得k张成的kN形成一个类群代数。我们确定了在|N| = |B| -1的情况下，环扩展kN∧A的结构。

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.