On the computation of fusion over the affine Temperley–Lieb algebra

Jonathan Belletête,Yvan Saint-Aubin

doi:10.1016/j.nuclphysb.2018.10.016

Abstract

Fusion product originates in the algebraization of the operator product expansion in conformal field theory. Read and Saleur (2007) introduced an analogue of fusion for modules over associative algebras, for example those appearing in the description of 2d lattice models. The article extends their definition for modules over the affine Temperley–Lieb algebra TLna.Since the regular Temperley–Lieb algebra TLn is a subalgebra of the affine TLna, there is a natural pair of adjoint induction-restriction functors (↑ar,↓ra). The existence of an algebra morphism ϕ:TLna→TLn provides a second pair of adjoint functors (⇑ar,⇓ar). Two fusion products between TLa-modules are proposed and studied. They are expressed in terms of these four functors. The action of these functors is computed on the standard, cell and irreducible TLna-modules. As a byproduct, the Peirce decomposition of TLna(q+q−1), when q is not a root of unity, is given as direct sum of the induction ↑raSn,k of standard TLn-modules to TLna-modules. Examples of fusion products of various pairs of affine modules are given.

Highlights

Since the regular Temperley-Lieb algebra TLn is a subalgebra of the affine TLan, there is a natural pair of adjoint inductionrestriction functors (↑ar, ↓ar )
The operator product of local boundary fields is sometimes seen as bringing points on the boundary together
This article introduced two fusion products over the affine Temperley-Lieb algebra through the two morphisms TLn →ι TLan and TLan →φ TLn

Summary

THE TEMPERLEY-LIEB ALGEBRAS

This section defines the fusion functors to be considered in this paper. It uses the langage of categories introduced in the study of the affine Temperley-Lieb family by Graham and Lehrer [7]. The affine Temperley-Lieb algebra TLan(q) is defined as the C-algebra of endomorphisms of n in TLa(q) It is the C-algebra generated by e0, τ, and τ−1 (that is, en,0, τn and τn−1). Our goal is to define, and compute, similar fusion products for the affine Temperley-Lieb category. It is relatively straightforward to show that (− ×1f −)m,n defines a commutative and associative fusion product on the affine Temperley-Lieb category. It is commutative because the fusion functors on TL are, and it is associative because ⇑ra ◦ ⇓ra −→∼ id modTLn and the fusion product on TL is associative. The bifunctors (− ×2f −)m,n defines a commutative and associative fusion product on the affine Temperley-Lieb category, for the same reasons. We chose to limit ourselves here to the two fusion products (− ×1f −) and (− ×2f −)

MODULES AND MORPHISMS

RESTRICTION AND INDUCTION FUNCTORS

CONCLUDING REMARKS