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The double sign of a real division algebra of finite dimension greater than one
Author(s) -
Darpö Erik,
Dieterich Ernst
Publication year - 2012
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201100189
Subject(s) - mathematics , division (mathematics) , sign (mathematics) , multiplication (music) , dimension (graph theory) , invariant (physics) , combinatorics , algebra over a field , pure mathematics , division algebra , arithmetic , mathematical analysis , filtered algebra , mathematical physics
For any real division algebra A of finite dimension greater than one, the signs of the determinants of left multiplication and right multiplication by an element a ∈ A ∖{0} are shown to form an invariant of A , called its double sign. For each n ∈ {2, 4, 8}, the double sign causes the category \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {D}_n$\end{document} of all n ‐dimensional real division algebras to decompose into four blocks. The structures of these blocks are closely related, and their relationship is made precise for a sample of full subcategories of \documentclass{article}\usepackage{amssymb,mathrsfs}\begin{document}\pagestyle{empty}$\mathscr {D}_n$\end{document} .