  The Temperley-Lieb algebra \tln(\beta) can be defined as the set of
rectangular diagrams with n points on each of their vertical sides, with all
points joined pairwise by non-intersecting strings. The multiplication is then
the concatenation of diagrams. The dilute Temperley-Lieb algebra \dtl n(\beta)
has a similar diagrammatic definition where, now, points on the sides may
remain free of strings. Like \tl n, the dilute \dtl n depends on a parameter
\beta\in\mathbb C, often given as \beta=q+q^{-1} for some q\in\mathbb C^\times.
In statistical physics, the algebra plays a central role in the study of dilute
loop models. The paper is devoted to the construction of its principal
indecomposable modules.
  Basic definitions and properties are first given: the dimension of \dtl n,
its break up into even and odd subalgebras and its filtration through n+1
ideals. The standard modules \U{n,k} are then introduced and their behaviour
under restriction and induction is described. A bilinear form, the Gram
product, is used to identify their (unique) maximal submodule \dr{n,k} which is
then shown to be irreducible or trivial. It is then noted that \dtl n is a
cellular algebra. This fact allows for the identification of complete sets of
non-isomorphic irreducible modules and projective indecomposable ones. The
structure of \dtl n as a left module over itself is then given for all values
of the parameter q, that is, for both q generic and a root of unity.
