1. CMB Online first
 Banica, Teodor

Tannakian duality for affine homogeneous spaces
Associated to any closed quantum subgroup $G\subset U_N^+$ and
any index set $I\subset\{1,\dots,N\}$ is a certain homogeneous
space $X_{G,I}\subset S^{N1}_{\mathbb C,+}$, called affine homogeneous
space. We discuss here the abstract axiomatization of the algebraic
manifolds $X\subset S^{N1}_{\mathbb C,+}$ which can appear in
this way, by using Tannakian duality methods.
Keywords:quantum isometry, noncommutative manifold Categories:46L65, 46L89 

2. CMB Online first
3. CMB 2014 (vol 57 pp. 708)
 Brannan, Michael

Strong Asymptotic Freeness for Free Orthogonal Quantum Groups
It is known that the normalized standard generators of the free
orthogonal quantum group $O_N^+$ converge in distribution to a free
semicircular system as $N \to \infty$. In this note, we
substantially improve this convergence result by proving that, in
addition to distributional convergence, the operator norm of any
noncommutative polynomial in the normalized standard generators of
$O_N^+$ converges as $N \to \infty$ to the operator norm of the
corresponding noncommutative polynomial in a standard free
semicircular system. Analogous strong convergence results are obtained
for the generators of free unitary quantum groups. As applications of
these results, we obtain a matrixcoefficient version of our strong
convergence theorem, and we recover a well known $L^2$$L^\infty$ norm
equivalence for noncommutative polynomials in free semicircular
systems.
Keywords:quantum groups, free probability, asymptotic free independence, strong convergence, property of rapid decay Categories:46L54, 20G42, 46L65 

4. CMB 2011 (vol 55 pp. 260)
 Delvaux, L.; Van Daele, A.; Wang, Shuanhong

A Note on the Antipode for Algebraic Quantum Groups
Recently, Beattie, Bulacu ,and Torrecillas proved Radford's formula for the fourth power of the antipode for a coFrobenius Hopf algebra.
In this note, we show that this formula can be proved for any regular multiplier Hopf
algebra with integrals (algebraic quantum groups). This, of course, not only includes the case of a
finitedimensional Hopf algebra, but also that of any
Hopf algebra with integrals (coFrobenius Hopf algebras). Moreover, it turns out that
the proof in this more general situation, in fact, follows in a few lines from wellknown formulas obtained earlier in the
theory of regular multiplier Hopf algebras with integrals.
We discuss these formulas and their importance in this theory. We also mention their generalizations, in particular to the (in a certain sense) more general theory of locally compact quantum groups. Doing so, and also because the proof of the main result itself is very short, the present note becomes largely of an expository nature.
Keywords:multiplier Hopf algebras, algebraic quantum groups, the antipode Categories:16W30, 46L65 
