This manuscript is devoted to the study of the concept of a generating subset (a.k.a. Hopf image of a morphism) in the setting of locally compact quantum groups. The aim of this paper is to provide an accurate description of the Hopf image of a given morphism. We extend and unify the previously existing approaches for compact and discrete quantum groups and present some results that can shed light on some local perspective in the theory of quantum groups. In particular, we provide a characterization of fullness of Hopf image in the language of partial actions as well as in representation-theoretic terms in the spirit of representation C * -categories, extending some known results not only to the broader setting of non-compact quantum groups, but also encompassing a broader setting of generating subsets.2010 Mathematics Subject Classification. Primary: 46L89 Secondary: 46L85, 46L52.

Abstract. In this note we refine the notion of conditionally negative definite kernels to the notion of conditionally strictly negative definite kernels and study its properties. We show that the class of these kernels carries some surprising rigidity, in particular, the word metric function on Coxeter groups is conditionally strictly negative definite if and only if the group is a free product of a number of copies of Z / 2Z 's and that the class of conditionally strictly negative definite kernels on a finite set is a one-parameter perturbation of the class of strictly positive definite kernels on this set. We also discuss several examples.

Abstract. Motivated by a question of A. Skalski and P.M. Sołtan ( ) about inner faithfulness of the S. Curran's map of extending a quantum increasing sequence to a quantum permutation, we revisit the results and techniques of T. Banica and J. Bichon ( ) and study some group-theoretic properties of the quantum permutation group on points. is enables us not only to answer the aforementioned question in positive in case n = , k = , but also to classify the automorphisms of S + , describe all the embeddings O − ( ) ⊂ S + and show that all the copies of O − ( ) inside S + are conjugate. We then use these results to show that the converse to the criterion we applied to answer the aforementioned question is not valid.