LAUSR

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 TL n a . Since the regular Temperley–Lieb algebra TL n is a subalgebra of the affine TL n a , there is a natural pair of adjoint induction-restriction functors ( ↑ a r , ↓ r a ) . The existence of an algebra morphism ϕ : TL n a → TL n provides a second pair of adjoint functors ( ⇑ a r , ⇓ a r ) . Two fusion products between TL a -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 TL n a -modules. As a byproduct, the Peirce decomposition of TL n a ( q + q − 1 ) , when q is not a root of unity, is given as direct sum of the induction ↑ r a S n , k of standard TL n -modules to TL n a -modules. Examples of fusion products of various pairs of affine modules are given.