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By J. L. Dupont, I. H. Madsen

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Example text

Mit Hilfe der orthogonalen Multiplikationen auf ee, II und 0 haben wir gezeigt, daB Sl, S3 und S7 parallelisierbar sind. ren sind. Aus diesem Satz folgt, daB es keine Dim~msionen n =f:. 1,2,4,8 gibt, fiir die orthogonale Multiplikationen yom Typ (n, n, n) existieren. = = = 37 2 Allgemeine Topologie Der Anschaulichkeit halber haben wir uns im vorigen Kapitel durchweg auf die Betrachtung von Teilraumen des IRn beschrankt. Es zeigt sich jedoch, daB dieser Ansatz nicht flexibel genug ist: Viele Konstruktionen, die man gerne durchfiihren wiirde, lassen sich so gar nicht oder nur mit miihsamem Aufwand realisieren.

Sie muB daher konstant sein, so daB

Dies geschieht in mehreren Schritten, die jeweils elementar zu verifizieren sind: 1. Die Elemente von JR kommutieren mit allen Oktaven. Es sind auch die einzigen Oktaven mit dieser Eigenschaft. 2. Fur x, y E 4) gilt xy = yx . 3. Fur x E 4) gilt xx = xx = IIxll 2 E JR. 4. Fur x,y E 4) gilt x(xy) = (xx)y. - Wegen 2. ist dann auch die Gleichung (yx)x = y(xx) fUr aIle x, y E 4) erfUllt. 5. Fur x, y E 4) und r E JR sind Produkte aus x, y, r assoziativ. - Dies laBt sich am besten so einsehen: Sei Ass( a, b, c) := (ab)c - a(bc).

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