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

TL;DR

This paper extends the concept of fusion for modules over the affine Temperley-Lieb algebra, introducing new fusion products, computing their actions on various modules, and analyzing the algebra's structure via induction functors.

Contribution

It introduces and studies two new fusion products for affine Temperley-Lieb modules, expanding the algebra's module theory and providing explicit computations and structural decompositions.

Findings

Defined two fusion products for affine Temperley-Lieb modules.

Computed the action of induction and restriction functors on standard, cell, and irreducible modules.

Provided examples of fusion products and the Peirce decomposition of the algebra.

Abstract

Fusion product originates in the algebraisation 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 $\atl n$. Since the regular Temperley-Lieb algebra $\tl n$ is a subalgebra of the affine $\atl n$, there is a natural pair of adjoint induction-restriction functors $(\Indar{}, \Resar{})$. The existence of an algebra morphism $\phi:\atl n\to\tl n$ provides a second pair of adjoint functors $(\Indphi{},\Resphi{})$. Two fusion products between $\atl{}$-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 $\atl n$-modules. As a byproduct, the Peirce decomposition of $\atl n(q+q^{-1})$, when $q$ is not a root of unity, is given as direct sum of the induction $\Indar{\TheS{n,k}}$ of standard $\tl n$-modules to $\atl n$-modules. Examples of fusion products of various pairs of affine modules are given.