Group-like algebras and Hadamard matrices

We give a description in terms of square matrices of the family of group-like algebras with S∗id=id∗S=uε. In the case that S=id and k⊆R, this translation take us to Hadamard matrices and, particularly, to examples of bi-Frobenius algebras satisfying S∗id=id∗S=uε 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 bi-Frobenius algebras with S∗id=id∗S=uε, presenting a version of Maschke's theorem for this family.