A model for normed algebras in rational G-spectra

TL;DR

This paper develops a simplified model for rational G-spectra and classifies normed algebras within this framework, extending previous results to a broader class of algebraic structures.

Contribution

It introduces a new simplified model for rational G-spectra and provides a classification of I-normed algebras, generalizing prior work by Wimmer.

Findings

Constructed a simplified model for rational G-spectra.

Classified I-normed algebras as collections of commutative algebras with compatible morphisms.

Extended Wimmer's classification to a broader setting.

Abstract

For a finite group $G$, we construct a simplified model for the $G$-symmetric monoidal $G$-$\infty$-category of rational $G$-spectra. Using this model, we classify $\mathcal{I}$-normed algebras in rational $G$-spectra for a given indexing system $\mathcal{I}$. We show that such an algebra is equivalently described as a collection $\{\mathcal{X}(G/H)\}_{(H\leq G)}$ of commutative algebras in nonequivariant rational spectra, indexed by conjugacy classes of subgroups of $G$, together with compatible morphisms of commutative algebras $\mathcal{X}(G/K)\xrightarrow{}\mathcal{X}(G/H)$ whenever $K\leq H$ and the induced map $G/K\xrightarrow{}G/H$ is in $\mathcal{I}$. This generalizes a result by Wimmer arXiv:1905.12420.