Categorical Aspects of Compact Quantum Groups

Applied Categorical Structures (Impact Factor: 0.69). 08/2012; 23(3). DOI: 10.1007/s10485-013-9333-8
Source: arXiv


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.

Full-text preview

Available from:
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: The category of locally compact quantum groups can be described as either Hopf *-homomorphisms between universal quantum groups, or as bicharacters on reduced quantum groups. We show how S{\o}ltan's quantum Bohr compactification can be used to construct a "compactification" in this category. Depending on the viewpoint, different C$^*$-algebraic compact quantum groups are produced, but the underlying Hopf *-algebras are always, canonically, the same. We show that a complicated range of behaviours, with C$^*$-completions between the reduced and universal level, can occur even in the cocommutative case, thus answering a question of S{\o}ltan. We also study such compactifications from the perspective of (almost) periodic functions. We give a definition of a periodic element in $L^\infty(\G)$, involving the antipode, which allows one to compute the Hopf *-algebra of the compactification of $\G$; we later study when the antipode assumption can be dropped. In the cocommutative case we make a detailed study of Runde's notion of a completely almost periodic functional-- with a slightly strengthening, we show that for [SIN] groups this does recover the Bohr compactification of $\hat G$.
    Preview · Article · Jul 2013 · Illinois journal of mathematics
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We show that provided $n\ne 3$, the involutive Hopf *-algebra $A_u(n)$ coacting universally on an $n$-dimensional Hilbert space has enough finite-dimensional representations in the sense that every non-zero element acts non-trivially in some finite-dimensional *-representation. This implies that the discrete quantum group with group algebra $A_u(n)$ is maximal almost periodic (i.e. it embeds in its quantum Bohr compactification), answering a question posed by P. So tan. We also prove analogous results for the involutive Hopf *-algebra $B_u(n)$ coacting universally on an $n$-dimensional Hilbert space equipped with a non-degenerate bilinear form.
    Preview · Article · Oct 2014 · Journal of Functional Analysis