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

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 (up arrow(a)(r), down arrow(a)(r)). The existence of an algebra morphism phi: TLna -> TLn provides a second pair of adjoint functors (double up arrow(r)(a), double down arrow(r)(a)). 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 up arrow( a)(r) S-n,S-k of standard TLn-modules to TLna-modules. Examples of fusion products of various pairs of affine modules are given. (C) 2018 The Author(s). Published by Elsevier B.V.