By Daniel B. Shapiro
The imperative subject of this booklet is the concept of Hurwitz and Radon touching on composition formulation for sums of squares, first proved within the 1920's. ideas from algebra and topology are used to generalize that theorem in numerous instructions. The textual content contains labored examples and lots of routines which increase nonetheless extra diversifications of the principal subject. the most viewers is those who have had a few graduate classes in summary algebra, yet many sections of the booklet are available to an individual with a few education in linear algebra. numerous significant issues during this e-book can be of curiosity to scholars of topology and geometry. the writer has tried to make the presentation as transparent and as straightforward as attainable
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Extra info for Compositions of quadratic forms
See Exercise 26. Exercises for Chapter 1 1. Composition algebras. Suppose S ⊆ Sim(V , q) with dim S = dim V . 1) of the appendix. (Hint. Scale q to assume it represents 1 and choose e ∈ V with q(e) = 1. Identify S with V by f → f (e). 9) provides a pairing. ) 26 1. Spaces of Similarities 2. Let (V , q) and (W, q ) be two quadratic spaces. Define the “adjoint” map J = V : Hom(V , W ) → Hom(W, V ). JW V is J W . How does J behave on matrices? 9). (3) σ < Sim(q, q ) if and only if q< Sim(σ, q ). 3.
22 1. Spaces of Similarities We will determine all the possible composition algebras over F . The classical examples over the field R include the real numbers (dim = 1), the complex numbers (dim = 2), the quaternions (dim = 4) and the octonions (dim = 8). We assume no knowledge of these examples and consider an arbitrary composition algebra A over F . This appendix is organized as a sequence of numbered statements, with hints for their proofs. 2. [ac, ad] = [a] · [c, d]. (Set b = c + d in (∗)).
Similarities f and g are comparable iff the angle between f (v) and g(v) is constant, independent of v. 4. (1) Suppose q 1, a . If f ∈ Sim• (q) then either f = x y −ay x or f = x ay for some x, y ∈ F with µ(f ) = x 2 + ay 2 . 8. If q 1, −1 , list all the 0-similarities. be the quaternion algebra with norm form q = 1, 1, 1, 1 . (2) Let D = −1,−1 F Let L0 = L(D) ⊆ Sim(D, q) arising from the left regular representation of D. a b c d d −c −b a Then L0 is the set of all . Similarly R0 = R(D) is the set −c −d a b −d c −b a a b c d −b a −d c of all .