Julien Bichon
Blaise Pascal University
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Publication
Featured researches published by Julien Bichon.
Communications in Mathematical Physics | 2006
Julien Bichon; An De Rijdt; Stefaan Vaes
We construct new examples of ergodic coactions of compact quantum groups, in which the multiplicity of an irreducible corepresentation can be strictly larger than the dimension of the latter. These examples are obtained using a bijective correspondence between certain ergodic coactions on C*-algebras and unitary fiber functors on the representation category of a compact quantum group. We classify these unitary fiber functors on the universal orthogonal and unitary quantum groups. The associated C*-algebras and von Neumann algebras can be defined by generators and relations, but are not yet well understood.
arXiv: Quantum Algebra | 2003
Julien Bichon
A quantum analogue of the automorphism group of a finite graph is introduced. These are quantum subgroups of the quantum permutation groups defined by Wang. The quantum automorphism group is a stronger invariant for finite graphs than the usual automorphism group. We get a quantum dihedral group D 4 .
Communications in Algebra | 2003
Julien Bichon
Abstract We show that the representation category of the quantum group of a non-degenerate bilinear form is monoidally equivalent to the representation category of the quantum group SL q (2) for a well chosen non-zero parameter q. The key ingredient for the proof of this result is the direct and explicit construction of an appropriate Hopf bigalois extension. Then we get, when the base field is of characteristic zero, a full description of cosemisimple Hopf algebras whose representation semi-ring is isomorphic to the one of SL(2).
Crelle's Journal | 2009
Teodor Banica; Julien Bichon
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.
Journal of Pure and Applied Algebra | 2001
Julien Bichon
Abstract A sovereign monoidal category is an autonomous monoidal category endowed with the choice of an autonomous structure and an isomorphism of monoidal functors between the associated left and right duality functors. In this paper we define and study the algebraic counterpart of sovereign monoidal categories: cosovereign Hopf algebras. We describe the universal cosovereign Hopf algebras, we study finite-dimensional cosovereign Hopf algebras via the dimension theory provided by the sovereign structure and we examine an example generalizing the quantum groups SL q .
Glasgow Mathematical Journal | 2010
Teodor Banica; Julien Bichon
We develop a general theory of Hopf image of a Hopf algebra representation, with the associated concept of inner faithful representation, modelled on the notion of faithful representation of a discrete group. We study several examples, including group algebras, enveloping algebras of Lie algebras, pointed Hopf algebras, function algebras, twistings and cotwistings, and we present a Tannaka duality formulation of the notion of Hopf image.
Asian-european Journal of Mathematics | 2008
Julien Bichon
We discuss some algebraic aspects of quantum permutation groups, working over arbitrary fields. If
Journal of The Institute of Mathematics of Jussieu | 2007
Teodor Banica; Julien Bichon
K
Journal of Algebra | 2003
Julien Bichon
is any characteristic zero field, we show that there exists a universal cosemisimple Hopf algebra coacting on the diagonal algebra
Journal of Algebra and Its Applications | 2006
Julien Bichon
K^n