Division algebras graded by a finite group

Abstract

Let G be a finite group and D a division algebra faithfully G-graded, finite dimensional over its center K, where char(K)=0. Let e∈G denote the identity element and suppose K0=K∩De, the e-center of D, contains ζnG, a primitive nG-th root of unity, where nG 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 Hom(M(H),μnH)G/H. Here μnH denotes the group of nH-th roots of unity, nH=exp(H), and M(H) is the Schur multiplier of H. The action of G/H on μnH is trivial and the action on M(H) is induced by the action of G on H.Our main theorem is the converse: Given an extension 1→H→G→Q→1, where H is abelian, a positive integer d, and an element of Hom(M(H),μnH)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 algebraically closed field of characteristic zero that admit a division algebra form whose e-center contains μnG.