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Search: MSC category 20G42 ( Quantum groups (quantized function algebras) and their representations [See also 16T20, 17B37, 81R50] )

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1. CMB Online first

Józiak, Paweł
 Remarks on Hopf images and quantum permutation groups $S_n^+$ Motivated by a question of A. Skalski and P.M. SoÅtan (2016) about inner faithfulness of the S. Curran's map of extending a quantum increasing sequence to a quantum permutation, we revisit the results and techniques of T. Banica and J. Bichon (2009) and study some group-theoretic properties of the quantum permutation group on $4$ points. This enables us not only to answer the aforementioned question in positive in case $n=4, k=2$, but also to classify the automorphisms of $S_4^+$, describe all the embeddings $O_{-1}(2)\subset S_4^+$ and show that all the copies of $O_{-1}(2)$ inside $S_4^+$ are conjugate. We then use these results to show that the converse to the criterion we applied to answer the aforementioned question is not valid. Keywords:Hopf image, quantum permutation group, compact quantum groupCategories:20G42, 81R50, 46L89, 16W35

2. CMB 2015 (vol 58 pp. 233)

Bergen, Jeffrey
 Affine Actions of $U_q(sl(2))$ on Polynomial Rings We classify the affine actions of $U_q(sl(2))$ on commutative polynomial rings in $m \ge 1$ variables. We show that, up to scalar multiplication, there are two possible actions. In addition, for each action, the subring of invariants is a polynomial ring in either $m$ or $m-1$ variables, depending upon whether $q$ is or is not a root of $1$. Keywords:skew derivation, quantum group, invariantsCategories:16T20, 17B37, 20G42

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 non-commutative polynomial in the normalized standard generators of $O_N^+$ converges as $N \to \infty$ to the operator norm of the corresponding non-commutative 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 matrix-coefficient version of our strong convergence theorem, and we recover a well known $L^2$-$L^\infty$ norm equivalence for non-commutative polynomials in free semicircular systems. Keywords:quantum groups, free probability, asymptotic free independence, strong convergence, property of rapid decayCategories:46L54, 20G42, 46L65

4. CMB 2012 (vol 57 pp. 424)

Sołtan, Piotr M.; Viselter, Ami
 A Note on Amenability of Locally Compact Quantum Groups In this short note we introduce a notion called quantum injectivity'' of locally compact quantum groups, and prove that it is equivalent to amenability of the dual. Particularly, this provides a new characterization of amenability of locally compact groups. Keywords:amenability, conditional expectation, injectivity, locally compact quantum group, quantum injectivityCategories:20G42, 22D25, 46L89
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