  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 $Gal(\bar{F}/F) \to PGL_n(C)$ lift to $GL_n(C)$. We
take special interest in the interaction of this result with algebraicity (on
the automorphic side) and geometricity (in the sense of Fontaine-Mazur). On the
motivic side, we study 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.
