  We consider the W-extended logarithmic minimal model WLM(p,p'). As in the
rational minimal models, the so-called fundamental fusion algebra of WLM(p,p')
is described by a simple graph fusion algebra. The fusion matrices in the
regular representation thereof are mutually commuting, but in general not
diagonalizable. Nevertheless, we show that they can be brought simultaneously
to block-diagonal forms whose blocks are upper-triangular matrices of dimension
1, 3, 5 or 9. The directed graphs associated with the two fundamental modules
are described in detail. The corresponding adjacency matrices share a complete
set of common generalized eigenvectors organized as a web constructed by
interlacing the Jordan chains of the two matrices. This web is here called a
Jordan web and it consists of connected subwebs with 1, 3, 5 or 9 generalized
eigenvectors. The similarity matrix, formed by concatenating these vectors,
simultaneously brings the two fundamental adjacency matrices to Jordan
canonical form modulo permutation similarity. The ranks of the participating
Jordan blocks are 1 or 3, and the corresponding eigenvalues are given by
2cos(j\pi/n) where j=0,...,n and n=p,p'. For p>1, only some of the modules in
the fundamental fusion algebra of WLM(p,p') are associated with boundary
conditions within our lattice approach. The regular representation of the
corresponding fusion subalgebra has features similar to the ones in the regular
representation of the fundamental fusion algebra, but with dimensions of the
upper-triangular blocks and connected Jordan-web components given by 1, 2, 3 or
8. Some of the key results are illustrated for W-extended critical percolation
WLM(2,3).
