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Division algebras graded by a finite group. (English) Zbl 07335734

Summary: Let \(G\) be a finite group and \(D\) a division algebra faithfully \(G\)-graded, finite dimensional over its center \(K\), where \(c h a r(K) = 0\). Let \(e\in G\) denote the identity element and suppose \(K_0=K\cap D_e\), the \(e\)-center of \(D\), contains \(\zeta_{n_G}\), a primitive \(n_G\)-th root of unity, where \(n_G\) 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 \(Hom(M(H),\mu_{n_H})^{G/H}\). Here \(\mu_{n_H}\) denotes the group of \(n_H\)-th roots of unity, \(n_H=exp(H)\), and \(M(H)\) is the Schur multiplier of \(H\). The action of \(G/H\) on \(\mu_{n_H}\) 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 \(1\to H\to G\to Q\to 1\), where \(H\) is abelian, a positive integer \(d\), and an element of \(Hom(M(H),\mu_{n_H})^Q\), 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 \(\mu_{n_G}\).

MSC:

20-XX Group theory and generalizations
16-XX Associative rings and algebras
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