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.

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In this paper we show that all complex semisimple Hopf algebras of dimension less than 24 are compact quantum groups. To do this, we survey all the above algebras and show explicitly that they can be described by bicrossed products of group algebras and its duals. We also study the behaviour under twisting of compact quantum groups. Using this we show that certain families of triangular semisimple Hopf algebras are compact quantum groups.

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.

We define the concept of companion automorphism of a Hopf algebra H as an automorphism σ : H Ñ H: σ 2 " S 2 -where S denotes the antipode-. A Hopf algebra is said to be almost involutive (AI) if it admits a companion automorphism that can be viewed as a special additional symmetry. We present examples and study some of the basic properties and constructions of AI-Hopf algebras centering the attention in the finite dimensional case. In particular we show that within the family of Hopf algebras of dimension smaller or equal than 15, only in dimension eight and twelve, there are non almost involutive Hopf algebras.