U-duality and M-theory, an algebraic approach

Based on our work hep-th/9809039, we discuss how U-duality arises as an exact symmetry of M-theory from T-duality and 11D diffeomorphism invariance. A set of Weyl generators are shown to realize the Weyl group of SO(d,d,Z) and E_{d(d)}(Z), while Borel generators extend these finite groups into the full T- and U-duality groups. We discuss how the BPS states fall into various representations, and obtain duality invariant mass formulae, relevant for the computation of exact string amplitudes. The realization of U-duality symmetry in Matrix gauge theory is also considered.