Indecomposable Involution Algebras
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An indecomposable involution algebra is one which cannot be written as a product of two involution algebras. The reason for looking at these is twofold. Firstly certain results are best stated in these terms and many proofs reduce partly or completely to this case, in particular of course all those which only involve the algebra, not the order. Secondly indecomposable involution algebras provide explicit illustrations for general result, indicating how these extend known classical ones, and allowing explicit computations. The study of the concrete situations arising here leads by necessity back from the language of Hom-groups to a more traditional one of groups of elements or ideles in the centre, e.g., replacing our determinants by reduced norms. This has the advantage that the reader can interpret the general theory in terms more familiar to him. It will become apparent that in this reinterpretation a variety of distinct cases have to be considered separately. This exhibits once more the power of our formalism of Hom groups, determinants and Pfaffians to provide a unified language for all these different situations. E.g. our discriminant generalises various invariants for particular indecomposable involution algebras, which occur in the literature.
KeywordsExact Sequence Division Algebra Maximal Order Open Subgroup Matrix Ring
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