I will present ongoing work on representations of the Temperley-Lieb algebra arising in connection with the $U_q(sl_2)$ invariant XXZ spin-chain when the deformation parameter $q$ lies on the unit circle. Using concepts from “non-Hermitian quantum mechanics” (namely PT-symmetry, Bender’s C-operator and quasi-Hermiticity) I will review the procedure known as “quantum group reduction” and present some exact results concerning the construction of an inner product which is invariant under the action of the Temperley-Lieb algebra. For a particular section of the unit circle I will present a novel formula for the invariant product which can be evaluated using Kauffman diagrams.

Department of Mathematics, University of Glasgow