## Abstract

Let G be a finite group and D a division algebra faithfully G-graded, finite dimensional over its center K, where . Let denote the identity element and suppose , the e-center of D, contains , a primitive -th root of unity, where is the exponent of G. To such a G-grading on D we associate a normal abelian subgroup H of G, a positive integer d and an element of . Here denotes the group of -th roots of unity, , and is the Schur multiplier of H. The action of on is trivial and the action on is induced by the action of G on H.

Our main theorem is the converse: Given an extension , where H is abelian, a positive integer d, and an element of , there is a division algebra as above that realizes these data. We apply this result to classify the G-graded simple algebras whose e-center is an algebraically closed field of characteristic zero that admit a division algebra form whose e-center contains .

Our main theorem is the converse: Given an extension , where H is abelian, a positive integer d, and an element of , there is a division algebra as above that realizes these data. We apply this result to classify the G-graded simple algebras whose e-center is an algebraically closed field of characteristic zero that admit a division algebra form whose e-center contains .

Original language | American English |
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Pages (from-to) | 1-25 |

Journal | Journal of Algebra |

Volume | 579 |

DOIs | |

State | Submitted - 1 Aug 2021 |

## Keywords

- Graded algebras
- Graded division algebras
- Division algebra which are G-graded
- G-graded simple algebras
- k-forms of algebras
- Twisted forms