Compact Kac algebras and commuting squares

We consider commuting squares of finite dimensional von Neumann algebras having the algebra of complex numbers in the lower left corner. Examples include the vertex models, the spin models (in the sense of subfactor theory) and the commuting squares associated to finite dimensional Kac algebras. To any such commuting square we associate a compact Kac algebra and we compute the corresponding subfactor and its standard invariant in terms of it.Comment: 14 pages, some minor change

Hopf algebras and subfactors associated to vertex models

If H is a Hopf algebra whose square of the antipode is the identity, v\in\l (V)\otimes H is a corepresentation, and \pi :H\to\l (W) is a representation, then $u=(id\otimes\pi)v$ satisfies the equation $(t\otimes id)u^{-1}=((t\otimes id)u)^{-1}$ of the vertex models for subfactors. A universal construction shows that any solution $u$ of this equatio n arises in this way. A more elaborate construction shows that there exists a ``minimal'' triple $(H,v,\pi)$ satisfying $(id\otimes\pi)v=u$. This paper is devoted to the study of this latter construction of Hopf algebras. If $u$ is unitary we construct a \c^*-norm on $H$ and we find a new description of the standard invariant of the subfactor associated to $u$. We discuss also the ``twisted'' (i.e. $S^2\neq id$) case.Comment: 25 pages, Late

Higher transitive quantum groups: theory and models

We investigate the notion of $k$-transitivity for the quantum permutation groups $G\subset S_N^+$, with a brief review of the known $k=1,2$ results, and with a study of what happens at $k\geq3$. We discuss then matrix modelling questions for the algebras $C(G)$, notably by introducing the related notions of double and triple flat matrix model. At the level of the examples, our main results concern the quantum groups coming from the complex Hadamard matrices, and from the Weyl matrices.Comment: 13 page

Quantum automorphism groups of homogeneous graphs

Associated to a finite graph $X$ is its quantum automorphism group $G$. The main problem is to compute the Poincar\'e series of $G$, meaning the series $f(z)=1+c_1z+c_2z^2+...$ whose coefficients are multiplicities of 1 into tensor powers of the fundamental representation. In this paper we find a duality between certain quantum groups and planar algebras, which leads to a planar algebra formulation of the problem. Together with some other results, this gives $f$ for all homogeneous graphs having 8 vertices or less.Comment: 30 page

Quantum groups and Fuss-Catalan algebras

The categories of representations of compact quantum groups of automorphisms of certain inclusions of finite dimensional C*-algebras are shown to be isomorphic to the categories of Fuss-Catalan diagrams.Comment: 12 page