On dualization of double frobenius algebras

Double Frobenius algebras (or dF-algebras) were recently introduced by the author. The concept generalizes finite-dimensional Hopf algebras, adjacency algebras of (non-commutative) association schemes, and C-algebras (or character algebras). In this paper we define a dualization construction of a dF-algebra, the so-called linear dual. We show that in the case of a Hopf algebra the linear dual gives the usual dual Hopf algebra and in the case of a C-algebra it essentially gives the usual Kawada’s dual.