A Calabi-Yau algebra with $E_6$ symmetry and the Clebsch-Gordan series of $sl(3)$
N. Crampe, L. Poulain d'Andecy, L. Vinet
TL;DR
The paper builds a Calabi–Yau three-generator algebra $\\mathcal{A}^{gen}$ and shows it surjects onto the diagonal centraliser $Z_2(sl(3))$ in $U(sl(3))^{\otimes 2) }$, with a central element forcing a natural quotient. Specialising to triples of $sl(3)$ highest weights reveals an $E_6$ Weyl group action on the weight parameters, and the specialised centraliser $Z_2(sl(3))^{spec}$ is described entirely in terms of the fundamental $E_6$ invariants. A concrete realisation in $U(sl(3))\otimes U(sl(3))$ exhibits the degrees of the central terms as the first five $E_6$ invariants, and a degree-12 Casimir-like relation closes the presentation. The work connects to spherical subalgebras of symplectic reflection algebras of type $E_6$ and links to Racah/Hahn algebras via Heun-type realizations, suggesting rich symmetry and representation-theoretic structure for the Clebsch–Gordan problem of $sl(3)$. It opens avenues for addressing missing-label problems, potential Bethe-ansatz approaches, and broader generalisations to other Lie algebras and embedding patterns.
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.