Integrable boundary conditions and W-extended fusion in the logarithmic minimal models LM(1,p)

We consider the logarithmic minimal models as 'rational' logarithmic conformal field theories with extended symmetry. To make contact with the extended picture starting from the lattice, we identify 4p - 2 boundary conditions as specific limits of integrable boundary conditions of the underlying Yang-Baxter integrable lattice models. Specifically, we identify 2p integrable boundary conditions to match the 2p known irreducible -representations. These 2p extended representations naturally decompose into infinite sums of the irreducible Virasoro representations (r, s). A further 2p - 2 reducible yet indecomposable -representations of rank 2 are generated by fusion and these decompose as infinite sums of indecomposable rank-2 Virasoro representations. The fusion rules in the extended picture are deduced from the known fusion rules for the Virasoro representations of and are found to be in agreement with previous works. The closure of the fusion algebra on a finite number of representations in the extended picture is remarkable confirmation of the consistency of the lattice approach