A Temperley-Lieb (TL) loop model is a Yang-Baxter integrable lattice model
with nonlocal degrees of freedom. On a strip of width N, the evolution operator
is the double-row transfer tangle D(u), an element of the TL algebra TL_N(beta)
with loop fugacity beta=2cos(lambda). Similarly on a cylinder, the single-row
transfer tangle T(u) is an element of the enlarged periodic TL algebra. The
logarithmic minimal models LM(p,p') comprise a subfamily of the TL loop models
for which the crossing parameter lambda=(p'-p)pi/p' is parameterised by coprime
integers 0<p<p'. For these special values, additional symmetries allow for
particular degeneracies in the spectra that account for the logarithmic nature
of these theories. For critical dense polymers LM(1,2), D(u) and T(u) are known
to satisfy inversion identities that allow us to obtain exact eigenvalues in
any representation and for all system sizes N. The generalisation for p'>2
takes the form of functional relations for D(u) and T(u) of polynomial degree
p'. These derive from fusion hierarchies of commuting transfer tangles
D^{m,n}(u) and T^{m,n}(u) where D(u)=D^{1,1}(u) and T(u)=T^{1,1}(u). The fused
transfer tangles are constructed from (m,n)-fused face operators involving
Wenzl-Jones projectors P_k on k=m or k=n nodes. Some projectors P_k are
singular for k>p'-1, but we argue that D^{m,n}(u) and T^{m,n}(u) are well
defined for all m,n. For generic lambda, we derive the fusion hierarchies and
the associated T- and Y-systems. For the logarithmic theories, the closure of
the fusion hierarchies at n=p' translates into functional relations of
polynomial degree p' for D^{m,1}(u) and T^{m,1}(u). We also derive the closure
of the Y-systems for the logarithmic theories. The T- and Y-systems are the key
to exact integrability and we observe that the underlying structure of these
functional equations relate to Dynkin diagrams of affine Lie algebras.Comment: 77 page
