L'algebre des symetries quantiques d'Ocneanu et la classification des systemes conformes a 2D

TL;DR

This paper explores the algebraic structures of quantum symmetries in 2D conformal systems, introducing Weak Hopf Algebras and their applications to classify and compute partition functions, with explicit examples for su(2) and su(3) cases.

Contribution

It presents a new realization of Ocneanu's quantum symmetry algebra as a quotient of graph algebras, enabling simplified computation of conformal partition functions and generalization to higher su(n) cases.

Findings

Explicit construction of Ocneanu algebra for A3 diagram

Development of a simple algorithm for partition function determination

Extension of methods to su(n) cases beyond su(2)

Abstract

The partition functions of a 2D conformal system - the modular invariant one or the generalized ones, coming from the introduction of defect lines - are expressed in terms of a set of coefficients that have the particularity to form nimreps of certain algebras. These coefficients define the various structure maps of a new class of Hopf Algebras, called Weak Hopf Algebras, and can be encoded in a set of graphs. The aim of chapter 1 is a presentation of the actual knowledge on this subject. In chapter 2 the Weak Hopf Algebra together with its structures is introduced, in particular the Ocneanu algebra of quantum symmetries, that play a key role in the study of 2d conformal systems. We analyze in details these structures for the A3 diagram, associated to the affine su(2) conformal system. In chapter 3, we present a realization of the Ocneanu algebra of quantum symmetries, constructed as a quotient of the tensor square of a graph algebra (ADE graph for the affine (2) model). This realization allows obtaining a very simple algorithm for the determination of the partition functions associated with the conformal model. Our construction can be naturally generalized to the affine su(n) cases, with n > 2, where few results were known. All the cases related of su(2)-type as well as three particular examples of su(3)-type are explicitly treated in chapter 4.