Some constructions of compact quantum groups

Abstract

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.

Some advances about the existence of compact involutions in semisimple Hopf algebras

“…If H is a * -Hopf algebra and we consider in H ∨ the involution given by formula (33), then we can define a * -Hopf structure in D(H ) by (αx) * = x * α * , for all α ∈ H ∨ , x ∈ H . The following result is Corollary 4.4 in [3].…”

“…There are several ways to define a CQG, see for instance [40] for the original definition, which was extended and developed in [6] or [15]. See also [1] for an approach from the coalgebraic point of view, or [3] for other equivalent definitions in finite dimension. In any case a finite-dimensional CQG is always semisimple (see Prop.…”

Almost involutive Hopf algebras

“…This assertion follows easily form the fact that the antipode for the bicrossed product is simply Spaxq " S H pxqS A paq for a P A, x P H -see for example [2] or [6] for a detailed description of the structure of A ' H-.…”

“…In particular, if H is almost involutive, so is its Drinfel'd double DpHq. This follows from the fact that the Drinfel'd double can be viewed as a bicrossed product (see [2] or [6]) DpHq " H _cop ' H where the structure of matched pair is given by…”