Variations on a theorem of tate

Let F be a number field. These notes explore Galois-theoretic, automorphic, and motivic analogues and refinements of Tate's basic result that continuous projective representations \mathrm{Gal}(\overline{F}/F) \to \mathrm{PGL}_n(\mathbb{C}) lift to \mathrm{GL}_n(\mathbb{C}). The author takes special interest in the interaction of this result with algebraicity (for automorphic representations) and geometricity (in the sense of Fontaine-Mazur). On the motivic side, the author studies refinements and generalizations of the classical Kuga-Satake construction. Some auxiliary results touch on: possible infinity-types of algebraic automorphic representations; comparison of the automorphic and Galois "Tannakian formalisms" monodromy (independence-of-\ell) questions for abstract Galois representations.