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

TL;DR

This paper develops a foundational framework for representation theory over the sphere spectrum, introducing new concepts like $S$-Lie algebras and constructing a novel stable homotopy invariant for links related to $sl_k$.

Contribution

It establishes the theory of $S$-Lie algebras and their representations, and constructs a new stable homotopy type invariant for links, extending classical representation theory into stable homotopy theory.

Findings

Defined $S$-Lie algebras and their representations.

Constructed a Khovanov $sl_k$-stable homotopy type.

Provided a new link invariant using stable homotopy methods.

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.