In this article -that has also the intention to survey some known results in the theory of compact quantum groups using methods different from the standard and with a strong algebraic flavor-we consider compact•-coalgebras and Hopf algebras. In the case of a •-Hopf algebra we present a proof of the characterization of the compactness in terms of the existence of a positive definite integral, and use our methods to give an elementary proof of the uniqueness -up to conjugation by an automorphism of Hopf algebrasof the compact involution appearing in [4]. We study the basic properties of the positive square root of the antipode square that is a Hopf algebra automorphism that we call the positive antipode. We use it -as well as the unitary antipode and Nakayama automorphism-in order to enhance our understanding of the antipode itself.

We study the action of monads on categories equipped with several monoidal structures. We identify the structure and conditions that guarantee that the higher monoidal structure is inherited by the category of algebras over the monad. Monoidal monads and comonoidal monads appear as the base cases in this hierarchy. Monads acting on duoidal categories constitute the next case. We cover the general case of n-monoidal categories and discuss several naturally occurring examples in which n ≤ 3.

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.

We prove that a profinite algebra whose left (right) cyclic modules are torsionless is finite dimensional and QF. We give a relative version of the notion of left (right) PF ring for pseudocompact algebras and prove it is left-right symmetric and dual to the notion of quasi-co-Frobenius coalgebras. We also prove two ring theoretic conjectures of Faith, in the setting (and supplementary hypothesis) of profinite algebras: any profinite semiartinian selfinjective algebra is finite dimensional and QF, and any FGF profinite algebra is finite dimensional QF.2000 Mathematics Subject Classification. 16T15, 16D50, 16D99.

Abstract. The purpose of this paper is to consider some basic constructions in the category of compact quantum groups -for example the case of extensions-with special emphasis in the finite dimensional situation. We give conditions, in some cases necessary and sufficient, to extend to the new objects the original compact structure.