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 Mosc. Math. J., 2002, Volume 2, Number 1, Pages 41–80 (Mi mmj45)

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.

DOI: https://doi.org/10.17323/1609-4514-2002-2-1-41-80

Full text: http://www.ams.org/.../abst2-1-2002.html
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Bibliographic databases:

MSC: 81R50, 81R05, 81T40, 82B20, 46L37
Received: February 7, 2001; in revised form December 20, 2001
Language:

Citation: R. Coquereaux, “Notes on the quantum tetrahedron”, Mosc. Math. J., 2:1 (2002), 41–80

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Kirillov A., Ostrik V., “On a $q$-analogue of the McKay correspondence and the ADE classification of $\mathfrak{sl}_2$ conformal field theories”, Adv. Math., 171:2 (2002), 183–227
2. Coquereaux R., Huerta M., “Torus structure on graphs and twisted partition functions for minimal and affine models”, J. Geom. Phys., 48:4 (2003), 580–634
3. Coquereaux R., Schieber G., “Determination of quantum symmetries for higher $ADE$ systems from the modular $T$ matrix”, J. Math. Phys., 44:9 (2003), 3809–3837
4. Chui C.H.O., Mercat Ch., Pearce P.A., “Integrable and conformal twisted boundary conditions for $\mathrm{sl}(2)$ $A$-$D$-$E$ lattice models”, J. Phys. A, 36:11 (2003), 2623–2662
5. Coquereaux R., “Quantum geometry of ADE diagrams and generalized Coxeter-Dynkin systems”, GROUP 24: Physical and Mathematical Aspects of Symmetries, Proceedings of the 24th International Colloquium on Group Theoretical Methods in Physics (Paris, 15–20 July 2002), Institute of Physics Conference Series, 173, 2003, 61–71
6. Coquereaux R., Isasi E., “On Quantum Symmetries of the Non-Ade Graph F-4”, Adv. Theor. Math. Phys., 8:6 (2004), 955–985
7. Coquereaux R., Trinchero R., “On Quantum Symmetries of Ade Graphs”, Adv. Theor. Math. Phys., 8:1 (2004), 189–216
8. Hammaoui D., Schieber G., Tahri E.H., “Higher Coxeter graphs associated with affine su(3) modular invariants”, J. Phys. A, 38:38 (2005), 8259–8286
9. Coquereaux R., “The A(2) Ocneanu quantum groupoid”, Algebraic Structures and Their Representations, Contemporary Mathematics Series, 376, 2005, 227–247
10. Coquereaux R., Hammaoui D., Schieber G., Tahri E.H., “Comments about quantum symmetries of SU(3) graphs”, J. Geom. Phys., 57:1 (2006), 269–292
11. Trinchero R., “Quantum symmetries of face models and the double triangle algebra”, Adv. Theor. Math. Phys., 10:1 (2006), 49–75
12. Isasi E., Schieber G., “From modular invariants to graphs: the modular splitting method”, J. Phys. A, 40:24 (2007), 6513–6537
13. Coquereaux R., Schieber G., “Orders and dimensions for $\mathrm{sl}(2)$ or $\mathrm{sl}(3)$ module categories and boundary conformal field theories on a torus”, J. Math. Phys., 48:4 (2007), 043511, 17 pp.
14. Coquereaux R., “Racah-Wigner quantum $6j$ symbols, Ocneanu cells for $A_N$ diagrams and quantum groupoids”, J. Geom. Phys., 57:2 (2007), 387–434
15. Hammaoui D., “The smallest Ocneanu quantum groupopd of SU(3) type”, Arab. J. Sci. Eng. Sect. C Theme Issues, 33:2 (2008), 225–238
16. Coquereaux R., Schieber G., “From conformal embeddings to quantum symmetries: an exceptional SU(4) example”, International Conference on Noncommutative Geometry and Physics, Journal of Physics Conference Series, 103, 2008
17. Robert Coquereaux, Gil Schieber, “Quantum Symmetries for Exceptional $\mathrm{SU}(4)$ Modular Invariants Associated with Conformal Embeddings”, SIGMA, 5 (2009), 044, 31 pp.
18. Goto S., “On Ocneanu's theory of double triangle algebras for subfactors and classification of irreducible connections on the Dynkin diagrams”, Expo Math, 28:3 (2010), 218–253