On quantum symmetries of ADE graphs
Robert Coquereaux (1), Roberto Trinchero (2) ((1) CPT - Marseille,, (2) CAB - Bariloche)
TL;DR
This paper explores the quantum symmetries of ADE graphs by analyzing the double triangle algebra's structure, reconstructing its bialgebra properties, and demonstrating it as a weak *-Hopf algebra, with detailed example of A3.
Contribution
It provides a reconstruction-based description of the DTA's bialgebra structure and proves it forms a weak *-Hopf algebra, advancing understanding of quantum symmetries in ADE graphs.
Findings
DTA has a bialgebra structure based on representation theory
DTA can be characterized as a weak *-Hopf algebra
Detailed analysis of the A3 graph case
Abstract
The double triangle algebra(DTA) associated to an ADE graph is considered. A description of its bialgebra structure based on a reconstruction approach is given. This approach takes as initial data the representation theory of the DTA as given by Ocneanu's cell calculus. It is also proved that the resulting DTA has the structure of a weak *-Hopf algebra. As an illustrative example, the case of the graph A3 is described in detail.
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