Hopf Algebras and Subfactors Associated to Vertex Models

Hopf Images and Inner Faithful Representations

“…In this section, we study Hopf images for various function algebras: polynomial functions on algebraic groups and representative functions on compact groups. The idea for the computation of the Hopf image goes back to [6], but we are a little bit more general here. These simple examples are already interesting for testing the possibility of generalizing representation theoretic properties of discrete group algebras to arbitrary Hopf algebras.…”

“…The aim of this paper is to provide an axiomatization and systematic study of the concept of Hopf image of a Hopf algebra representation, as well as the related concept of inner faithful representation. These notions appeared, under various degrees of generality, in a number of independent investigations: vertex models and related quantum groups [6,11,12], locally compact quantum groups and their outer actions [20,27]. These were used extensively in the recent paper [10] in order to study the quantum symmetries of Hadamard matrices and of the corresponding subfactors.…”

Compact Kac Algebras and Commuting Squares

“…We denote by ✷ u the vertex model associated to a biunitary u. We recall from [2] that if H is a Hopf * -algebra, v ∈ M n ⊗ H is a unitary corepresentation and π : H → M k is a * -representation then the element (id ⊗π)v ∈ M n ⊗M k is a biunitary. Conversely, given a biunitary u ∈ M n ⊗ M k , the category of triples (H, v, π) such that (id ⊗ π)v = u has a universal object, called the minimal model for u.…”

“…It follows that there exists a representation ν : A aut (B) → L(A) such that (id ⊗ ν)V = u ϕ . Thus u ϕ is biunitary and (A aut (B), V, ν) is a model for it (see [2]).…”

On the structure of quantum permutation groups

“…An element x in a non-commutative C * -probability space is called semicircular if its spectral measure is dµ x (t) = (2π) 2], and 0 elsewhere.…”