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

Abstract

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 modules and their duals, Harish-Chandra pairs and Zuckermann functors.

As an application, we construct a Khovanov sl_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.

Keywords

• Homotopy categorification
• Khovanov homology
• Spectral algebra
• Spectral representation theory