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Linearly reductive quantum groups: descent, simplicity and finiteness properties
 Chirvasitu, Alexandru Leonida
 Advisor(s): Serganova, Vera
Abstract
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 finitedimensional 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 nondegeneracy conditions. Several natural families of compact quantum groups, some with noncommutative 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 ndimensional Hilbert space has enough finitedimensional representations, in the sense that every nonzero element acts nontrivially in some finitedimensional *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 ndimensional Hilbert space equipped with a nondegenerate bilinear form.
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