Division algebras graded by a finite group

Eli Aljadeff, Darrell Haile, Yaakov Karasik

Research output: Contribution to journalArticlepeer-review

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 .
Original languageAmerican English
Pages (from-to)1-25
JournalJournal of Algebra
Volume579
DOIs
StateSubmitted - 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

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