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Notes on the quantum tetrahedron
R. Coquereaux CNRS – Center of Theoretical Physics
Abstract:
This is a set of notes describing several aspects of the space of paths on ADE Dynkin diagrams, with a particular attention paid to the graph
$E_6$. Many results originally due to A. Ocneanu are described here in a very elementary way (manipulation of square or rectangular matrices). We recall the concept of essential matrices (intertwiners) for a graph and describe their module properties with respect to right and left actions of fusion algebras. In the case of the graph
$E_6$, essential matrices build up a right module with respect to its own fusion algebra, but a left module with respect to the fusion algebra of
$A_{11}$. We present two original results: 1) Our first contribution is to show how to recover the Ocneanu graph of quantum symmetries of the Dynkin diagram
$E_6$ from the natural multiplication defined in the tensor square of its fusion algebra (the tensor product should be taken over a particular subalgebra); this is the Cayley graph for the two generators of the twelve-dimensional algebra
$E_6\otimes_{A_3}E_6$ (here
$A_3$ and
$E_6$ refer to the commutative fusion algebras of the corresponding graphs). 2) To every point of the graph of quantum symmetries one can associate a particular matrix describing the “torus structure” of the chosen Dynkin diagram; following Ocneanu, one obtains in this way, in the case of
$E_6$, twelve such matrices of dimension
$11\times 11$, one of them is a modular invariant and encodes the partition function of which corresponding conformal field theory. Our own next contribution is to provide a simple algorithm for the determination of these matrices.
Key words and phrases:
ADE, conformal field theory, Platonic bodies, path algebras, subfactors, modular invariance, quantum groups, quantum symmetries, Racah–Wigner bigebra.
MSC: 81R50,
81R05,
81T40,
82B20,
46L37 Received: February 7, 2001; in revised form
December 20, 2001
Language: English
DOI:
10.17323/1609-4514-2002-2-1-41-80
Fulltext:
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