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Julien Bichon

Publications and source records attributed to Julien Bichon.

At least 55 records · Page 3Linked to original sources
arXiv2007

Algebraic quantum permutation groups

We discuss some algebraic aspects of quantum permutation groups, working over arbitrary fields. If $K$ is any characteristic zero field, we show that there exists a universal cosemisimple Hopf algebra coacting on the diagonal algebra $K^n$: this is a refinement of Wang's universality theorem for the (compact) quantum permutation group. We also prove a structural result for Hopf algebras having a non-ergodic coaction on the diagonal algebra $K^n$, on which we determine the possible group gradings when $K$ is algebraically closed and has characteristic zero.

math.QA↗
arXiv2007

Quantum groups acting on 4 points

We classify the compact quantum groups acting on 4 points. These are the quantum subgroups of the quantum permutation group $\mathcal Q_4$. Our main tool is a new presentation for the algebra $\rm C(\mathcal Q_4)$, corresponding to an isomorphism of type $\mathcal Q_4\simeq SO_{-1}(3)$. The quantum subgroups of $\mathcal Q_4$ are subject to a McKay type correspondence, that we describe at the level of algebraic invariants.

math.QA↗
arXiv2007

The hyperoctahedral quantum group

We consider the hypercube in $\mathbb R^n$, and show that its quantum symmetry group is a $q$-deformation of $O_n$ at $q=-1$. Then we consider the graph formed by $n$ segments, and show that its quantum symmetry group is free in some natural sense. This latter quantum group, denoted $H_n^+$, enlarges Wang's series $S_n^+,O_n^+,U_n^+$.

math.RT↗
arXiv2007

Quantum permutation groups: a survey

This is a presentation of recent work on quantum permutation groups. Contains: a short introduction to operator algebras and Hopf algebras; quantum permutation groups, and their basic properties; diagrams, integration formulae, asymptotic laws, matrix models; the hyperoctahedral quantum group, free wreath products, quantum automorphism groups of finite graphs, graphs having no quantum symmetry; complex Hadamard matrices, cocycle twists of the symmetric group, quantum groups acting on 4 points; remarks and comments.

math.CO↗
arXiv2006

Quantum automorphism groups of vertex-transitive graphs of order $\leq 11$

We study quantum automorphism groups of vertex-transitive graphs having less than 11 vertices. With one possible exception, these can be obtained from cyclic groups ${\mathbb Z}_n$, symmetric groups $S_n$ and quantum symmetric groups $\mathcal Q_n$, by using various product operations. The exceptional case is that of the Petersen graph, and we present some questions about it.

math.QA↗
arXiv2006

Graphs having no quantum symmetry

We consider circulant graphs having $p$ vertices, with $p$ prime. To any such graph we associate a certain number $k$, that we call type of the graph. We prove that for $p>>k$ the graph has no quantum symmetry, in the sense that the quantum automorphism group reduces to the classical automorphism group.

math.CO↗
arXiv2006

Free product formulae for quantum permutation groups

Associated to a finite graph $X$ is its quantum automorphism group $G(X)$. We prove a formula of type $G(X*Y)=G(X)*_wG(Y)$, where $*_w$ is a free wreath product. Then we discuss representation theory of free wreath products, with the conjectural formula $μ(G*_wH)=μ(G)\boxtimesμ(H)$, where $μ$ is the associated spectral measure. This is verified in two situations: one using free probability techniques, the other one using planar algebras.

math.QA↗
arXiv2005

Ergodic coactions with large multiplicity and monoidal equivalence of quantum groups

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.

math.OA↗
arXiv2004

Lazy cohomology: an analogue of the Schur multiplier for arbitrary Hopf algebras

We propose a detailed systematic study of a group H^2_L(A) associated, by elementary means of lazy 2-cocycles, to any Hopf algebra A. This group was introduced by Schauenburg (with a different name) in order to generalize G.I. Kac's exact sequence. We study the various interplays of lazy cohomology in Hopf algebra theory: Galois and biGalois objects, Brauer groups and projective representations. We obtain a Kac-Schauenburg-type sequence for double crossed products of possibly infinite-dimensional Hopf algebras. Finally the explicit computation of H^2_L(A) for monomial Hopf algebras and for a class of cotriangular Hopf algebras is performed.

math.QA↗
arXiv2004

Galois and biGalois objects over monomial non semisimple Hopf algebras

We describe the Galois objects and biGalois groups of monomial nonsemisimple Hopf algebras. The main feature of our description is the use of modified versions of the second cohomology group of the grouplike elements. These computations generalize the previous ones of Masuoka and Schauenburg for the Taft algebras.

math.QA↗
arXiv2002

Hopf-Galois Systems

We introduce the concept of Hopf-Galois system, a reformulation of the notion of Galois extension of the base field for a Hopf algebra. The main feature of our definition is a generalization of the antipode of an ordinary Hopf algebra. The main application of Hopf-Galois systems is the construction of monoidal equivalences between comodule categories over Hopf algebras.

math.QA↗
arXiv2001

The representation category of the quantum group of a non-degenerate bilinear form

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 main ingredient for the proof of this result is the direct and explicit construction of an appropriate Hopf bigalois extension. Then we get, in characteristic zero, a full description of cosemisimple Hopf algebras whose representation semi-ring is isomorphic to the one of SL(2).

math.QA↗
arXiv2001

Free wreath product by the quantum permutation group

A free wreath product construction of a Hopf algebra (or of a Woronowicz algebra) by Wang's quantum permutation group is done. It provides new examples of quantum groups and is useful to describe the quantum automorphism group of the n-times disjoint union of a finite connected graph.

math.QA↗
arXiv1999

Galois reconstruction of finite quantum groups

We describe a universal factorization for a functor with values in finite-dimensional measured algebras. More precisely we contruct the quantum automorphism group of this functor. This general recontruction result allows us to recapture a finite-dimensional Hopf algebra from the category of finite-dimensional measured comodule algebras.

math.QA↗
arXiv1999

Quantum automorphism groups of finite graphs

We determine the quantum automorphism groups of finite graphs. 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 one. We get a quantum dihedral group D4.

math.QA↗