Group-Like algebras and Hadamard matrices

Abstract

We give a description in terms of square matrices of the family of group-like algebras with $S\ast id=id\ast S=u\epsilon$. In the case that $S=id$ and $char\mathbb{k}$ is not 2 and does not divide the dimension of the algebra, this translation take us to Hadamard matrices and, particularly, to examples of biFrobenius algebras satisfying $S\ast id=id\ast S=u\epsilon$ and that are not Hopf algebras. Finally, we generalize some known results on separability and coseparability valid for finite dimensional Hopf algebras to this special class of biFrobenius algebras with $S\ast id=id\ast S=u\epsilon$, presenting a version of Maschke's theorem for this family.