  This paper focuses on recollements and silting theory in triangulated
categories. It consists of two main parts. In the first part a criterion for a
recollement of triangulated subcategories to lift to a torsion torsion-free
triple (TTF triple) of ambient triangulated categories with coproducts is
proved. As a consequence, lifting of TTF triples is possible for recollements
of stable categories of repetitive algebras or self-injective finite length
algebras and recollements of bounded derived categories of separated Noetherian
schemes. When, in addition, the outer subcategories in the recollement are
derived categories of small linear categories the conditions from the criterion
are sufficient to lift the recollement to a recollement of ambient triangulated
categories up to equivalence. In the second part we use these results to study
the problem of constructing silting sets in the central category of a
recollement generating the t-structure glued from the silting t-structures in
the outer categories. In the case of a recollement of bounded derived
categories of Artin algebras we provide an explicit construction for gluing
classical silting objects.
