Building on classical invariant theory, it is observed that the polarised
traces generate the centraliser ZLβ(sl(N)) of the diagonal embedding of
U(sl(N)) in U(sl(N))βL. The paper then focuses on sl(3) and the
case L=2. A Calabi--Yau algebra A with three generators is
introduced and explicitly shown to possess a PBW basis and a certain central
element. It is seen that Z2β(sl(3)) is isomorphic to a quotient of the
algebra 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
A arises, involving the pairs of numbers characterising the three
highest weights. In this realisation in U(sl(3))βU(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 E6β. With the correct association between the six parameters of the
representations and some roots of E6β, the symmetry under the full Weyl group
of type E6β is made manifest. The coefficients of the relations and the value
of the central element in the realisation in U(sl(3))βU(sl(3)) are
expressed in terms of the fundamental invariant polynomials associated to
E6β. It is also shown that the relations of the algebra A can be
realised with Heun type operators in the Racah or Hahn algebra.Comment: 24 page