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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, $gl_n(S)$-Verma modulesand their duals, Harish-Chandra pairs and Zuckermann functors. As anapplication, we construct a Khovanov $sl_k$-stable homotopy type with a largeprime hypothesis, which is a new link invariant, using a stable homotopyanalogue of the method of J.Sussan.

Derived representation theory of Lie algebras and stable homotopy categorification of $sl_k$