An algebraic model for rational toral G-spectra

TL;DR

This paper develops an algebraic model for the toral component of rational G-spectra, advancing the understanding of rational G-spectra for any compact Lie group G.

Contribution

It provides an algebraic model for the toral part of rational G-spectra, a key step toward modeling all rational G-spectra algebraically.

Findings

The abelian category from previous work models the toral G-spectra.

The homotopy category of rational toral G-spectra is a retract of all rational G-spectra.

This work advances the algebraic understanding of rational G-spectra for compact Lie groups.

Abstract

For G a compact Lie group, toral G-spectra are those rational G-spectra whose geometric isotropy consists of subgroups of a maximal torus of G. The homotopy category of rational toral G-spectra is a retract of the category of all rational G-spectra. In this paper we show that the abelian category defined by the second author in previous work gives an algebraic model for the toral part of rational G-spectra. This is a major step in establishing an algebraic model for all rational G-spectra, for any compact Lie group G.