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.