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.

26 pages

References in corpus (2)

• Calabi-Yau algebras
• Computing with matrix invariants

Cited by in corpus (8)

• The Askey-Wilson algebra and its avatars
• Heun operator of Lie type and the modified algebraic Bethe ansatz
• Polynomial algebras from $su(3)$ and the generic model on the two sphere
• Temperley-Lieb, Birman-Murakami-Wenzl and Askey-Wilson algebras and other centralizers of $U_q({\mathfrak{sl}}_2)$
• Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators
• Construction of polynomial algebras from intermediate Casimir invariants of Lie algebras
• Generalized quadratic commutator algebras of PBW-type
• Chern-Simons theory, link invariants and the Askey-Wilson algebra