  Virasoro Kac modules were initially introduced indirectly as representations
whose characters arise in the continuum scaling limits of certain transfer
matrices in logarithmic minimal models, described using Temperley-Lieb
algebras. The lattice transfer operators include seams on the boundary that use
Wenzl-Jones projectors. If the projectors are singular, the original
prescription is to select a subspace of the Temperley-Lieb modules on which the
action of the transfer operators is non-singular. However, this prescription
does not, in general, yield representations of the Temperley-Lieb algebras and
the Virasoro Kac modules have remained largely unidentified. Here, we introduce
the appropriate algebraic framework for the lattice analysis as a quotient of
the one-boundary Temperley-Lieb algebra. The corresponding standard modules are
introduced and examined using invariant bilinear forms and their Gram
determinants. The structures of the Virasoro Kac modules are inferred from
these results and are found to be given by finitely generated submodules of
Feigin-Fuchs modules. Additional evidence for this identification is obtained
by comparing the formalism of lattice fusion with the fusion rules of the
Virasoro Kac modules. These are obtained, at the character level, in complete
generality by applying a Verlinde-like formula and, at the module level, in
many explicit examples by applying the Nahm-Gaberdiel-Kausch fusion algorithm.
