  We construct an extension of f(T) gravity with the inclusion of a non-minimal
torsion-matter coupling in the action. The resulting theory is a novel
gravitational modification, since it is different from both f(T) gravity, as
well as from the non-minimal curvature-matter-coupled theory. The cosmological
application of this new theory proves to be very interesting. In particular, we
obtain an effective dark energy sector whose equation-of-state parameter can be
quintessence or phantom-like, or exhibit the phantom-divide crossing, while for
a large range of the model parameters the Universe results in a de Sitter,
dark-energy-dominated, accelerating phase. Additionally, we can obtain
early-time inflationary solutions too, and thus provide a unified description
of the cosmological history.

General Relativity and Quantum Cosmology;Cosmology and Nongalactic Astrophysics;High Energy Physics - Theory