Division algebras with a projective basis |
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Authors: | Eli Aljadeff Darrell Haile |
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Affiliation: | (1) Department of Mathematics, Technion-Israel Institute of Technology, 3200 Haifa, Israel;(2) Department of Mathematics, Indiana University, 47405 Bloomington, IN, USA |
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Abstract: | Letk be any field andG a finite group. Given a cohomology class α∈H 2(G,k *), whereG acts trivially onk *, one constructs the twisted group algebrak αG. Unlike the group algebrakG, the twisted group algebra may be a division algebra (e.g. symbol algebras, whereG⋞Z n×Zn). This paper has two main results: First we prove that ifD=k α G is a division algebra central overk (equivalentyD has a projectivek-basis) thenG is nilpotent andG’ the commutator subgroup ofG, is cyclic. Next we show that unless char(k)=0 and , the division algebraD=k α G is a product of cyclic algebras. Furthermore, ifD p is ap-primary factor ofD, thenD p is a product of cyclic algebras where all but possibly one are symbol algebras. If char(k)=0 and , the same result holds forD p, p odd. Ifp=2 we show thatD 2 is a product of quaternion algebras with (possibly) a crossed product algebra (L/k,β), Gal(L/k)⋞Z 2×Z2n. |
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