By Daniel B. Shapiro

The significant subject of this booklet is the concept of Hurwitz and Radon referring to composition formulation for sums of squares, first proved within the 1920's. innovations from algebra and topology are used to generalize that theorem in different instructions. The textual content comprises labored examples and plenty of routines, which advance nonetheless extra adaptations of the primary subject. the most viewers is those who have had a few graduate classes in summary algebra, yet many sections of the ebook are obtainable to somebody with a few education in linear algebra. a number of significant issues during this publication could be of curiosity to scholars of topology and geometry. the writer has tried to make the presen-tation as transparent and as uncomplicated as attainable.

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Example text

Replacing S by the isometric space f1−1 S we may assume that f1 = 1V . Then the equations above reduce to those given here. The converse follows similarly. The conditions above correspond to the second form of the Hurwitz Matrix Equations. With this formulation an experienced reader will notice that the algebra generated by the fi is related to the Clifford algebra C( −a2 , . . , −as ). This connection is explored in Chapter 4. 8 Example. Let q = 1, a with corresponding basis {e1 , e2 } of V .

Hint. (1) Show µ(f ) ∈ F •2 . (4) Assume F algebraically closed and f 2 = 1V . The eigenspaces U + and U − are totally isotropic of dimension m. Examine the matrix of g relative to V = U + ⊕ U − 0 1 using the Gram matrix . 1 0 (5) G ⊆ Sim+ (V ) by Wonenburger. Conversely it suffices to show that SO(q) ⊆ G. The maps τa generate O(q), where τa is the reflection fixing the hyperplane (a)⊥ . Therefore the maps τa τ1 generate SO(q). Writing [a] for q(a) as in the appendix, we −1 ¯ + a x)a ¯ = −[a]a xa.

Define f to be proper if det f = µ(f )m . The proper similarities form a subgroup Sim+ (V ) of index 2 in Sim• (V ). This is the analog of the special orthogonal group O+ (n) = SO(n). (3) Suppose f˜ = −f . If g = a1V + bf for a, b ∈ F then g is proper. (4) Wonenburger’s Theorem. Suppose f, g ∈ Sim• (V ) and f˜ = −f . If g commutes with f , then g is proper. If g anticommutes with f and 4 | n then g is proper. (5) Let L0 , R0 ⊆ Sim( 1, 1, 1, 1 ) be the subspaces described in Exercise 4(2). Let G be the group generated by L•0 and R0• .