Linearly reductive quantum groups: descent, simplicity and finiteness properties

The thesis comprises three largely independent projects undertaken during my stay at UC Berkeley, all revolving around the same mathematical objects: Cosemisimple Hopf algebras, regarded here as function algebras on linearly reductive quantum groups. We often specialize further to Hopf *-algebras coacting universally on finite-dimensional Hilbert spaces perhaps endowed with additional structure. Such a Hopf algebra is to be thought of as the algebra of representative functions on the compact quantum automorphism group of the respective structure. Chapter 2 is based on [23]. The question of whether or not a Hopf algebra H is faithfully flat over a Hopf subalgebra A has received positive answers in several particular cases: when H is commutative, or cocommutative, or pointed, or when A contains the coradical of H. We prove the result for cosemisimple H, adding this latter class of Hopf algebras to those known to be faithfully flat over all Hopf subalgebras. We also show that the third term of the resulting ``exact sequence'' A -> H -> C is always a cosemisimple coalgebra, and that the expectation H -> A is positive when H is a CQG algebra. Chapter 3 consists of material from [22], with earlier related results appearing in [24]. We define the notion of a (linearly reductive) center for a linearly reductive quantum group, and show that the quotient of a such a quantum group by its center is simple whenever its fusion semiring is free in the sense of Banica and Vergnioux. We also prove that the same is true of free products of quantum groups under very mild non-degeneracy conditions. Several natural families of compact quantum groups, some with non-commutative fusion semirings and hence very ``far from classical'', are thus seen to be simple. Examples include quotients of free unitary groups by their centers as well as quotients of quantum reflection groups by their centers. In Chapter 4 we show that provided n is different from 3, the involutive Hopf *-algebra Au(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 Au(n) is maximal almost periodic, i.e. it embeds in its quantum Bohr compactification; this answers a question posed by P. Soltan. We also prove analogous results for the involutive Hopf *-algebra Bu(n) coacting universally on an n-dimensional Hilbert space equipped with a non-degenerate bilinear form