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Mathematical Physics

arXiv:1812.08730 (math-ph)
[Submitted on 20 Dec 2018]

Title:The $SU(3)$ $A_1$ graph and its associated quantum groupoid

Authors:Maria Días, Esteban Isasi, Luis Vásquez
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Abstract:An explicit and complete construction of the $SU(3)$ $A_1$ associated quantum groupoid is presented in this work, inspired by the approach taken by Trinchero for the $SU(2)$ $A_l$ graphs. New creation and annihilation operators were defined in order to consider the $3$ different types of back-tracks which appear due to the specific structure of $SU(3)$. The $C^{\star}$ bialgebra and the realization of a Temperley-Lieb algebra is studied thoroughly. Finally, it is shown that the construction of the quantum groupoids associated to the $A_{1}$ $SU(N)$ graphs are easily obtained for any value of $N$ using the results of this work. The generalization for higher levels $A_l$ graphs are still an unsolved challenge, but now we count with enough tools, some insight about how to attack this problem, and the first steps towards solving it.
Comments: 25 pages (preprint version), 6 figures
Subjects: Mathematical Physics (math-ph)
MSC classes: 81R50
Report number: SB/F/480-18
Cite as: arXiv:1812.08730 [math-ph]
  (or arXiv:1812.08730v1 [math-ph] for this version)
  https://doi.org/10.48550/arXiv.1812.08730

Submission history

From: Esteban Isasi [view email]
[v1] Thu, 20 Dec 2018 18:01:28 UTC (23 KB)
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