Categorical Aspects of Compact Quantum Groups

Abstract

We show that either of the two reasonable choices for the category of compact quantum groups is nice enough to allow for a plethora of universal constructions, all obtained "by abstract nonsense" via the adjoint functor theorem. This approach both recovers constructions which have appeared in the literature, such as the quantum Bohr compactification of a locally compact semigroup, and provides new ones, such as the coproduct of a family of compact quantum groups, and the compact quantum group freely generated by a locally compact quantum space. In addition, we characterize epimorphisms and monomorphisms in the category of compact quantum groups.

Remarks on the quantum Bohr compactification

“…Indeed, if compactifications exist, then we have that C is "reflective" in B and the compactification is simply the "reflection" (see [33,Section IV.3]). This sort of "categorical" approach to defining the classical Bohr compactification of a group was stuided in [23,24], and for a similar treatment of the quantum case, see the recent paper [8] (which essentially gives a treatment of So ltan's work via abstract categorical arguments, but which does not consider the category LCQG described below).…”

Free Poisson Hopf algebras generated by coalgebras

“…The present paper is a contribution to the study of Poisson Hopf algebras from an algebraic point of view. More precisely, in light of the increasing interest shown recently in the category of Hopf algebras (see [2,3,4,5,8,17,18,19,23] and the references therein), we will study the categorical properties of Poisson Hopf algebras. As we will see, the category k-PoissBiAlg of Poisson bialgebras (and therefore the category of Poisson Hopf algebras) is not as friendly as the category k-BiAlg of bialgebras in the sense that it does not enjoy the same nice symmetry.…”