Fixed point algebras for easy quantum groups

Olivier Gabriel; Moritz Weber

doi:10.3842/SIGMA.2016.097

Fixed point algebras for easy quantum groups

Olivier Gabriel, Moritz Weber

Department of Mathematical Sciences

2 Citations (Scopus)

54 Downloads (Pure)

Abstract

Compact matrix quantum groups act naturally on Cuntz algebras. The first author isolated certain conditions under which the fixed point algebras under this action are Kirchberg algebras. Hence they are completely determined by their K-groups. Building on prior work by the second author,we prove that free easy quantum groups satisfy these conditions and we compute the K-groups of their fixed point algebras in a general form. We then turn to examples such as the quantum permutation group S⁺ _n,the free orthogonal quantum group O⁺ _n and the quantum reflection groups H^s+ _n. Our fixed point-algebra construction provides concrete examples of free actions of free orthogonal easy quantum groups,which are related to Hopf-Galois extensions.

Original language	English
Article number	097
Journal	Symmetry, Integrability and Geometry: Methods and Applications
Volume	12
Number of pages	21
ISSN	1815-0659
DOIs	https://doi.org/10.3842/SIGMA.2016.097
Publication status	Published - 2016

Keywords

Easy quantum groups
Free actions
Free orthogonal quantum groups
Fusion rules
K-theory
Kirchberg algebras
Noncrossing partitions
Quantum permutation groups
Quantum reflection groups

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title = "Fixed point algebras for easy quantum groups",

abstract = "Compact matrix quantum groups act naturally on Cuntz algebras. The first author isolated certain conditions under which the fixed point algebras under this action are Kirchberg algebras. Hence they are completely determined by their K-groups. Building on prior work by the second author,we prove that free easy quantum groups satisfy these conditions and we compute the K-groups of their fixed point algebras in a general form. We then turn to examples such as the quantum permutation group S+ n,the free orthogonal quantum group O+ n and the quantum reflection groups Hs+ n. Our fixed point-algebra construction provides concrete examples of free actions of free orthogonal easy quantum groups,which are related to Hopf-Galois extensions.",

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AB - Compact matrix quantum groups act naturally on Cuntz algebras. The first author isolated certain conditions under which the fixed point algebras under this action are Kirchberg algebras. Hence they are completely determined by their K-groups. Building on prior work by the second author,we prove that free easy quantum groups satisfy these conditions and we compute the K-groups of their fixed point algebras in a general form. We then turn to examples such as the quantum permutation group S+ n,the free orthogonal quantum group O+ n and the quantum reflection groups Hs+ n. Our fixed point-algebra construction provides concrete examples of free actions of free orthogonal easy quantum groups,which are related to Hopf-Galois extensions.

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KW - Free actions

KW - Free orthogonal quantum groups

KW - Fusion rules

KW - K-theory

KW - Kirchberg algebras

KW - Noncrossing partitions

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KW - Quantum reflection groups

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Fixed point algebras for easy quantum groups

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