Journal article

Publication year: 2021

Pages: 1-25

Journal: Journal of Algebra

Volume number: 579

ISSN: 0021-8693

eISSN: 1090-266X

DOI Link: https://doi.org/10.1016/j.jalgebra.2021.03.008

Publisher: Elsevier

Abstract: 

Let G be a finite group and Da division algebra faithfully G-graded, finite dimensional over its center K, where char(K) = 0. Let e is an element of G denote the identity element and suppose K-0= K boolean AND D-e, the e-center of D, contains.nG, a primitive nG-th root of unity, where nGis the exponentof G. To such a G-grading on Dwe associate a normal abelian subgroup Hof G, a positive integer dand an element of Hom(M(H), mu H-n)(G/H). Here mu nHdenotes the group of nH-th roots of unity, n(H)= exp(H), and M(H) is the Schur multiplier of H. The action of G/Hon mu nHis trivial and the action on M(H) is induced by the action of Gon H.

Our main theorem is the converse: Given an extension 1 -> H -> G -> Q -> 1, where His abelian, a positive integer d, and an element of Hom(M(H), mu nH)Q, 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 closed field of characteristic zero that admit a division algebra form whose e-center contains mu(nG). (C) 2021 Elsevier Inc. All rights reserved.

Harvard Citation style: Aljadeff, E., Haile, D. and Karasik, Y. (2021) Division algebras graded by afinite group, Journal of Algebra, 579, pp. 1-25. https://doi.org/10.1016/j.jalgebra.2021.03.008

APA Citation style: Aljadeff, E., Haile, D., & Karasik, Y. (2021). Division algebras graded by afinite group. Journal of Algebra. 579, 1-25. https://doi.org/10.1016/j.jalgebra.2021.03.008