  The interest in Logarithmic Conformal Field Theories (LCFTs) has been growing
over the last few years thanks to recent developments coming from various
approaches. A particularly fruitful point of view consists in considering
lattice models as regularizations for such quantum field theories. The
indecomposability then encountered in the representation theory of the
corresponding finite-dimensional associative algebras exactly mimics the
Virasoro indecomposable modules expected to arise in the continuum limit. In
this paper, we study in detail the so-called Temperley-Lieb (TL) fusion functor
introduced in physics by Read and Saleur [Nucl. Phys. B 777, 316 (2007)]. Using
quantum group results, we provide rigorous calculations of the fusion of
various TL modules. Our results are illustrated by many explicit examples
relevant for physics. We discuss how indecomposability arises in the "lattice"
fusion and compare the mechanisms involved with similar observations in the
corresponding field theory. We also discuss the physical meaning of our lattice
fusion rules in terms of indecomposable operator-product expansions of quantum
fields.
