An algebraic model for rational SO(3)-spectra

TL;DR

This paper advances the algebraic modeling of rational SO(3)-spectra by lifting known category equivalences to Quillen equivalences, paving the way for algebraic models of G-spectra for any compact Lie group G.

Contribution

It introduces a Quillen equivalence for rational SO(3)-spectra, extending Greenlees' category equivalence to a model category level, aiding future algebraic models for G-spectra.

Findings

Established a Quillen equivalence for rational SO(3)-spectra.

Provided methods for algebraic modeling of toral parts of rational G-spectra.

First step towards algebraic models for G-spectra of any compact Lie group.

Abstract

Greenlees established an equivalence of categories between the homotopy category of rational SO(3)-spectra and the derived category DA(SO(3)) of a certain abelian category. In this paper we lift this equivalence of homotopy categories to the level of Quillen equivalences of model categories. Methods used in this paper provide the first step towards obtaining an algebraic model for the toral part of rational G-spectra, for any compact Lie group G.