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We show that the discrete duals of the universal unitary quantum groups U+ n and orthogonal quantum groups O+ n have Kirchberg's factorization property when n ≠ 3. [ABSTRACT FROM AUTHOR]

Abstract: The notion of normal quantum subgroup introduced in algebraic context by Parshall and Wang when applied to compact quantum groups is shown to be equivalent to the notion of normal quantum subgroup introduced by the author. As applications, a quantum analog of the third fundamental isomorphism theorem for groups is obtained, which is used along with the equivalence theorem to obtain results on structure of quantum groups with property F and quantum groups with property FD. Other results on normal quantum subgroups for tensor products, free products and crossed products are also proved. [Copyright &y& Elsevier]

Abstract: The notion of simple compact quantum group is introduced. As non-trivial (noncommutative and noncocommutative) examples, the following families of compact quantum groups are shown to be simple: (a) The universal quantum groups for satisfying , ; (b) The quantum automorphism groups of finite-dimensional -algebras B endowed with the canonical trace τ when , including the quantum permutation groups on n points (); (c) The standard deformations of simple compact Lie groups K and their twists , as well as Rieffel''s deformation . [Copyright &y& Elsevier]

By introducing a result that guarantees a given bialgebra to be a Hopf algebra under a natural condition, we show that the quantum automorphism group of the algebra k[x] of polynomials over a field k (of any characteristic) is the universal quantum ax + b group A, generalizing the fact that the automorphism group of k[x] is the ax + b group. The q-deformation of the ax + b group is then seen as one among a certain family, A[sub q,n] (q ∈ k, n an integer) of quantum subgroups of this universal quantum group. [ABSTRACT FROM AUTHOR]

It is shown that the compact matrix quantum groups SUq(2) are non-isomorphic to each other for q-[-1, 1]-{0}, and that the compact matrix quantum groups SUq(n) are non-isomorphic to each other for q-(0, 1]. Some invariants for compact quantum groups are also discussed. [ABSTRACT FROM AUTHOR]