arXiv subjects
Publications and source records attributed to Julien Bichon.
We describe the quantum subspaces of Banica-Goswami's half-liberated real-spheres, showing in particular that there is a bijection between the symmetric ones and the conjugation stable closed subspaces of the complex projective spaces.
We study the discrete quantum groups $Γ$ whose group algebra has an inner faithful representation of type $π:C^*(Γ)\to M_K(\mathbb C)$. Such a representation can be thought of as coming from an embedding $Γ\subset U_K$. Our main result, concerning a certain class of examples of such quantum groups, is an asymptotic convergence theorem for the random walk on $Γ$. The proof uses various algebraic and probabilistic techniques.
We study the deformed tensor products of complex Hadamard matrices, $L_{ia,jb}=Q_{ib}H_{ij}K_{ab}$. One problem is that of reconstructing the quantum group $G_L\subset S_{NM}^+$ out of the quantum groups $G_H\subset S_N^+,G_K\subset S_M^+$ and of the parameter matrix $Q\in M_{N\times M}(\mathbb T)$, and we obtain here several results, including complete results in the Fourier matrix case, $H=F_N,K=F_M$, when the parameter matrix $Q$ is generic.
We construct, for q a root of unity of odd order, an embedding of the projective special linear group PSL(n) into the group of bi-Galois objects over u_q(sl(n))*, the coordinate algebra of the Frobenius-Lusztig kernel of SL(n), which is shown to be an isomorphism at n=2.
We discuss some axioms that ensure that a Hopf algebra has its simple comodules classified using an analogue of the Borel-Weil construction. More precisely we show that a Hopf algebra having a dense big cell satisfies the above requirement. This method has its roots in the work of Parshall and Wang in the case of q-deformed quantum groups GL and SL. Here we examine the example of universal cosovereign Hopf algebras, for which the weight group is the free group on two generators.
We study the (compact) quantum subgroups of the compact quantum group $SU_{-1}(3)$: we show that any non-classical such quantum subgroup is a twist of a compact subgroup of SU(3) or is isomorphic to a quantum subgroup of $U_{-1}(2)$.
We describe the universal quantum group preserving a preregular multilinear form, by means of an explicit finite presentation of the corresponding Hopf algebra.
We show that if $A$ and $H$ are Hopf algebras that have equivalent tensor categories of comodules, then one can transport what we call a free Yetter-Drinfeld resolution of the counit of $A$ to the same kind of resolution for the counit of $H$, exhibiting in this way strong links between the Hochschild homologies of $A$ and $H$. This enables us to get a finite free resolution of the counit of $\mathcal B(E)$, the Hopf algebra of the bilinear form associated to an invertible matrix $E$, generalizing an ealier construction of Collins, Hartel and Thom in the orthogonal case $E=I_n$. It follows that $\B(E)$ is smooth of dimension 3 and satisfies Poincaré duality. Combining this with results of Vergnioux, it also follows that when $E$ is an antisymetric matrix, the $L^2$-Betti numbers of the associated discrete quantum group all vanish. We also use our resolution to compute the bialgebra cohomology of $\B(E)$ in the cosemisimple case.
A half-commutative orthogonal Hopf algebra is a Hopf *-algebra generated by the self-adjoint coefficients of an orthogonal matrix corepresentation $v=(v_{ij})$ that half commute in the sense that $abc=cba$ for any $a,b,c \in \{v_{ij}\}$. The first non-trivial such Hopf algebras were discovered by Banica and Speicher. We propose a general procedure, based on a crossed product construction, that associates to a self-transpose compact subgroup $G \subset U_n$ a half-commutative orthogonal Hopf algebra $\mathcal A_*(G)$. It is shown that any half-commutative orthogonal Hopf algebra arises in this way. The fusion rules of $\mathcal A_*(G)$ are expressed in term of those of $G$.
We prove that the quantum group inclusion $O_n \subset O_n^*$ is "maximal", where $O_n$ is the usual orthogonal group and $O_n^*$ is the half-liberated orthogonal quantum group, in the sense that there is no intermediate compact quantum group $O_n\subset G\subset O_n^*$. In order to prove this result, we use: (1) the isomorphism of projective versions $PO_n^*\simeq PU_n$, (2) some maximality results for classical groups, obtained by using Lie algebras and some matrix tricks, and (3) a short five lemma for cosemisimple Hopf algebras.
We give examples of finite quantum permutation groups which arise from the twisting construction or as bicrossed products associated to exact factorizations in finite groups. We also give examples of finite quantum groups which are not quantum permutation groups: one such example occurs as a split abelian extension associated to the exact factorization $\mathbb S_4 = \mathbb Z_4 \mathbb S_3$ and has dimension 24. We show that, in fact, this is the smallest possible dimension that a non quantum permutation group can have.
We show that semisimple Hopf algebras having a self-dual faithful irreducible comodule of dimension 2 are always obtained as abelian extensions with quotient Z_2. We prove that nontrivial Hopf algebras arising in this way can be regarded as deformations of binary polyhedral groups and describe its category of representations. We also prove a strengthening of a result of Nichols and Richmond on cosemisimple Hopf algebras with a 2-dimensional irreducible comodule in the finite dimensional context. Finally, we give some applications to the classification of certain classes of semisimple Hopf algebras.
We prove that we have an isomorphism of type $A_{aut}(\mathbb C_σ[G])\simeq A_{aut}(\mathbb C[G])^σ$, for any finite group $G$, and any 2-cocycle $σ$ on $G$. In the particular case $G=\mathbb Z_n^2$, this leads to a Haar-measure preserving identification between the subalgebra of $A_o(n)$ generated by the variables $u_{ij}^2$, and the subalgebra of $A_s(n^2)$ generated by the variables $X_{ij}=\sum_{a,b=1}^np_{ia,jb}$. Since $u_{ij}$ is "free hyperspherical" and $X_{ij}$ is "free hypergeometric", we obtain in this way a new free probability formula, which at $n=\infty$ corresponds to the well-known relation between the semicircle law, and the free Poisson law.
We survey some aspects of the theory of Hopf-Galois objects that may studied advantageously by using the language of cogroupoids. These are the notes for a series of lectures given at Cordoba University, may 2010. The lectures are part of the course ``Hopf-Galois theory'' by Sonia Natale.
The notion of inner linear Hopf algebra is a generalization of the notion of discrete linear group. In this paper, we prove two general results that enable us to enlarge the class of Hopf algebras that are known to be inner linear: the first one is a characterization by using the Hopf dual, while the second one is a stability result under extensions. We also discuss the related notion of inner unitary Hopf *-algebra.
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.
We develop a combinatorial approach to the quantum permutation algebras, as Hopf images of representations of type $π:A_s(n)\to B(H)$. We discuss several general problems, including the commutativity and cocommutativity ones, the existence of tensor product or free wreath product decompositions, and the Tannakian aspects of the construction. The main motivation comes from the quantum invariants of the complex Hadamard matrices: we show here that, under suitable regularity assumptions, the computations can be performed up to $n=6$.
To any Hopf algebra H we associate two commutative Hopf algebras, which we call the first and second lazy homology Hopf algebras of H. These algebras are related to the lazy cohomology groups based on the so-called lazy cocycles of H by universal coefficient theorems. When H is a group algebra, then its lazy homology can be expressed in terms of the 1- and 2-homology of the group. When H is a cosemisimple Hopf algebra over an algebraically closed field of characteristic zero, then its first lazy homology is the Hopf algebra of the universal abelian grading group of the category of corepresentations of H. We also compute the lazy homology of the Sweedler algebra.