# Division algebras graded by a finite group

## TL;DR

This paper characterizes and constructs division algebras with finite group gradings over fields of characteristic zero, linking algebraic data to gradings and classifying certain G-simple algebras.

## Contribution

It provides a converse construction linking group gradings on division algebras to algebraic data, and classifies G-simple algebras admitting division algebra forms.

## Key findings

- Associates gradings with abelian subgroups, integers, and homomorphisms from Schur multipliers.
- Main theorem constructs division algebras from given algebraic data.
- Classifies G-simple algebras with division algebra forms over algebraically closed fields.

## Abstract

Let $k$ be a field containing an algebraically closed field of characteristic zero. If $G$ is a finite group and $D$ is a division algebra over $k$, finite dimensional over its center, we can associate to a faithful $G$-grading on $D$ a normal abelian subgroup $H$, a positive integer $d$ and an element of $Hom(M(H), k^\times)^G$, where $M(H)$ is the Schur multiplier of $H$. Our main theorem is the converse: Given an extension $1\rightarrow H\rightarrow G\rightarrow G/H\rightarrow 1$, where $H$ is abelian, a positive integer $d$, and an element of $Hom(M(H), k^\times)^G$, there is a division algebra with center containing $k$ that realizes these data. We apply this result to classify the $G$-simple algebras over an algebraically closed field of characteristic zero that admit a division algebra form over a field containing an algebraically closed field.