scholarly journals Towards a Classification of Compact Quantum Groups of Lie Type

2016 ◽  
pp. 231-263
Author(s):  
Sergey Neshveyev ◽  
Makoto Yamashita
Keyword(s):  
Quantum Groups ◽  
Compact Quantum Groups ◽  
Groups Of Lie Type

scholarly journals ON SECOND COHOMOLOGY OF DUALS OF COMPACT GROUPS

2011 ◽  
Vol 22 (09) ◽  
pp. 1231-1260 ◽  
Author(s):  
SERGEY NESHVEYEV ◽  
LARS TUSET
Keyword(s):  
Quantum Groups ◽  
Hopf Algebras ◽  
Cohomology Group ◽  
Tensor Category ◽  
Compact Groups ◽  
Tensor Categories ◽  
Connected Group ◽  
Compact Quantum Groups ◽  
Algebraic Counterpart

We show that for any compact connected group G the second cohomology group defined by unitary invariant two-cocycles on Ĝ is canonically isomorphic to [Formula: see text]. This implies that the group of autoequivalences of the C*-tensor category Rep G is isomorphic to [Formula: see text]. We also show that a compact connected group G is completely determined by Rep G. More generally, extending a result of Etingof–Gelaki and Izumi–Kosaki we describe all pairs of compact separable monoidally equivalent groups. The proofs rely on the theory of ergodic actions of compact groups developed by Landstad and Wassermann and on its algebraic counterpart developed by Etingof and Gelaki for the classification of triangular semisimple Hopf algebras. We give a self-contained account of amenability of tensor categories, fusion rings and discrete quantum groups, and prove an analog of Radford's theorem on minimal Hopf subalgebras of quasitriangular Hopf algebras for compact quantum groups.

scholarly journals Classification of Non-Kac Compact Quantum Groups of SU(n) Type

2015 ◽  
Vol 2016 (11) ◽  
pp. 3356-3391
Author(s):  
Sergey Neshveyev ◽  
Makoto Yamashita
Keyword(s):  
Quantum Groups ◽  
Compact Quantum Groups

scholarly journals RIESZ TRANSFORMS ON COMPACT QUANTUM GROUPS AND STRONG SOLIDITY

2021 ◽  
pp. 1-37
Author(s):  
Martijn Caspers
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Riesz Transform ◽  
Wreath Products ◽  
Compact Quantum Group ◽  
Riesz Transforms ◽  
Markov Semigroup ◽  
Markov Semigroups ◽  
Free Products ◽  
Compact Quantum Groups

Abstract One of the main aims of this paper is to give a large class of strongly solid compact quantum groups. We do this by using quantum Markov semigroups and noncommutative Riesz transforms. We introduce a property for quantum Markov semigroups of central multipliers on a compact quantum group which we shall call ‘approximate linearity with almost commuting intertwiners’. We show that this property is stable under free products, monoidal equivalence, free wreath products and dual quantum subgroups. Examples include in particular all the (higher-dimensional) free orthogonal easy quantum groups. We then show that a compact quantum group with a quantum Markov semigroup that is approximately linear with almost commuting intertwiners satisfies the immediately gradient- ${\mathcal {S}}_2$ condition from [10] and derive strong solidity results (following [10]). Using the noncommutative Riesz transform we also show that these quantum groups have the Akemann–Ostrand property; in particular, the same strong solidity results follow again (now following [27]).

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