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A Calabi-Yau algebra with $E_6$ symmetry and the Clebsch-Gordan series of $sl(3)$

Abstract : Building on classical invariant theory, it is observed that the polarised traces generate the centraliser $Z_L(sl(N))$ of the diagonal embedding of $U(sl(N))$ in $U(sl(N))^{\otimes L}$. The paper then focuses on $sl(3)$ and the case $L=2$. A Calabi--Yau algebra $\mathcal{A}$ with three generators is introduced and explicitly shown to possess a PBW basis and a certain central element. It is seen that $Z_2(sl(3))$ is isomorphic to a quotient of the algebra $\mathcal{A}$ by a single explicit relation fixing the value of the central element. Upon concentrating on three highest weight representations occurring in the Clebsch--Gordan series of $U(sl(3))$, a specialisation of $\mathcal{A}$ arises, involving the pairs of numbers characterising the three highest weights. In this realisation in $U(sl(3))\otimes U(sl(3))$, the coefficients in the defining relations and the value of the central element have degrees that correspond to the fundamental degrees of the Weyl group of type $E_6$. With the correct association between the six parameters of the representations and some roots of $E_6$, the symmetry under the full Weyl group of type $E_6$ is made manifest. The coefficients of the relations and the value of the central element in the realisation in $U(sl(3))\otimes U(sl(3))$ are expressed in terms of the fundamental invariant polynomials associated to $E_6$. It is also shown that the relations of the algebra $\mathcal{A}$ can be realised with Heun type operators in the Racah or Hahn algebra.
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Submitted on : Tuesday, June 16, 2020 - 10:04:00 PM
Last modification on : Tuesday, January 11, 2022 - 5:56:35 PM

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N. Crampe, L. Poulain d'Andecy, L. Vinet. A Calabi-Yau algebra with $E_6$ symmetry and the Clebsch-Gordan series of $sl(3)$. Journal of Lie Theory, Heldermann Verlag, In press. ⟨hal-02870776⟩

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