Extended T-systems, Q matrices and T-Q relations for sl(2) models at roots of unity

The mutually commuting 1 x n fused single and double-row transfer matrices of the critical six-vertex model are considered at roots of unity q = e(i lambda) with crossing parameter lambda = (p'-p)pi/p' a rational fraction of pi. The 1 x n transfer matrices of the dense loop model analogs, namely the logarithmic minimal models LM (p, p'), are similarly considered. For these sl(2) models, we find explicit closure relations for the T-system functional equations and obtain extended sets of bilinear T-system identities. We also define extended Q matrices as linear combinations of the fused transfer matrices and obtain extended matrix T-Q relations. These results hold for diagonal twisted boundary conditions on the cylinder as well as U-q(sl(2)) invariant/Kac vacuum and off-diagonal/Robin vacuum boundary conditions on the strip. Using our extended T-system and extended T-Q relations for eigenvalues, we deduce the usual scalar Baxter T-Q relation and the Bazhanov-Lukyanov- Zamolodchikov decomposition of the fused transfer matrices T-n(u + lambda) and D-n (u + lambda), at fusion level n = p' - 1, in terms of the product Q+ (u)Q(-)(u) or Q(u)(2). It follows that the zeros of TP'-1 (u + lambda) and Dp'-1(u + lambda) are comprised of the Bethe roots and complete p' strings. We also clarify the formal observations of Pronko and Yang-Nepomechie-Zhang and establish, under favourable conditions, the existence of an infinite fusion limit n -> infinity in the auxiliary space of the fused transfer matrices. Despite this connection, the infinite-dimensional oscillator representations are not needed at roots of unity due to finite closure of the functional equations.