Quantum groups acting on 4 points

By T. Banica, Julien Bichon. 2007

74 citations recorded by the source corpus.

Abstract We classify the compact quantum groups acting on 4 points. These are the quantum subgroups of the quantum permutation group 𝓠4. Our main tool is a new presentation for the algebra C(𝓠4), corresponding to an isomorphism of type 𝓠4 ≃ SO –1(3). The quantum subgroups of 𝓠4 are subject to a McKay type correspondence, that we describe at the level of algebraic invariants.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f6a4b22e6a35c2ad541064d6f93c5e329d606edd

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In Algebra and Number Theory

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Citations in the corpus

Reduced operator algebras of trace-preserving quantum automorphism groups (2012, background): “≥ 2 ( q−2 − q(2) − [2](4)q(1 + q (2))(2) [3]q[5]q − [2](2)q [4](2)q [3]q[5]q − 2[2](2)q(1 + q (2)) [5]q − 2[2]q [4]q[5]q − 2[2](3)q(1 + q (2)) [4]q[3]q[5]q ) ‖ξ‖(2) =: C(q)‖ξ‖(2),”
Quantum permutation groups: a survey (2006, background): “n , the compact quantum group dual to As(n), defined by As(n) = C(S + n ). At n = 4 we have the following result [ 6 ].”
Algebraic quantum permutation groups (2007, background): “then H is a Hopf ∗-algebra quotient of As(4, C) (listed in [ 4 ]) or is a Hopf ∗-algebra quotient of C(S3) ∗ C(S2).”
INVARIANTS OF THE HALF-LIBERATED ORTHOGONAL GROUP (2009, background): “These quantum groups allowed in particular a clarification of the relation with subfactors [3], noncommutative geometry [9] and free probability [15].”