Quantum Symmetry Groups of Finite Spaces

We determine the quantum automorphism groups of finite spaces and find they are all compact quantum groups in the sense of Woronowicz. This solves a problem of Connes for finite spaces.Comment: AMSLATEX, 21 pages. To appear in Commun Math Phy

On the problem of classifying simple compact quantum groups

We review the notion of simple compact quantum groups and examples, and discuss the problem of construction and classification of simple compact quantum groups. Several new quantum groups constructed by Banica, Curran and Speicher since the author's first paper on simple quantum groups are shown to be simple using results of Raum and Weber.Comment: 25 page

Rieffel Type Discrete Deformation of Finite Quantum Groups

We introduce a discrete deformation of Rieffel type for finite (quantum) groups. Using this, we give a non-trivial example of a finite quantum group of order 18. We also give a deformation of finite groups of Lie type by using their maximal abelian subgroups.Comment: AMSLATEX, 21 pages. Accepted for publication in Commun Math. Phys. Corrected an error occured at the end of the proof of Theorem 3.2, as well as some typos, as found by the refere