  In the context of Conformal Field Theory (CFT), many results can be obtained
from the representation theory of the Virasoro algebra. While the interest in
Logarithmic CFTs has been growing recently, the Virasoro representations
corresponding to these quantum field theories remain dauntingly complicated,
thus hindering our understanding of various critical phenomena. We extend in
this paper the construction of Read and Saleur (2007), and uncover a deep
relationship between the Virasoro algebra and a finite-dimensional algebra
characterizing the properties of two-dimensional statistical models, the
so-called blob algebra (a proper extension of the Temperley--Lieb algebra).
This allows us to explore vast classes of Virasoro representations (projective,
tilting, generalized staggered modules, etc.), and to conjecture a
classification of all possible indecomposable Virasoro modules (with, in
particular, L_0 Jordan cells of arbitrary rank) that may appear in a consistent
physical Logarithmic CFT where Virasoro is the maximal local chiral algebra. As
by-products, we solve and analyze algebraically quantum-group symmetric XXZ
spin chains and sl(2|1) supersymmetric spin chains with extra spins at the
boundary, together with the "mirror" spin chain introduced by Martin and
Woodcock (2004).
