Derived representation theory of Lie algebras and stable homotopy categorification of $s{l}_k$

We set up foundations of representation theory over $S$, the sphere spectrum, which is the `initial ring’ of stable homotopy theory. In particular, we treat $S$-Lie algebras and their representations, characters, $g{l}_n(S)$-Verma modules and their duals, Harish-Chandra pairs and Zuckermann functors. As an application, we construct a Khovanov $s{l}_k$-stable homotopy type with a large prime hypothesis, which is a new link invariant, using a stable homotopy analogue of the method of J.Sussan.