CHAPTER I

ALGEBRAIC THEORY OF QUADRATIC FORMS,

CLIFFORD ALGEBRAS, AN D SPIN GROUPS

1. Quadratic forms and associative algebras

1.1. Given vector spaces V and W over a field F, we let Hom(W, V; F) denote

the set of all F-linear maps of W into V. We then denote by End(V, F) the set

Hom(V, V; F) with an obvious ring-structure, and put GL(V, F) — End(V, F)x

and SL(V, F) = {ae GL(V, F) | det(a) = 1 } . We drop the letter F if that is

clear from the context. We let Hom(W, V) act on W on the right; namely we

denote by wa the image of w G W under a G Hom(W, V). If /3 G Hom(X, W)

with another vector space X, then the composite map of j3 and a can be written

pa, so that x(Pa) = {xp)a for x G X. The identity element of End(V) will often

be denoted by ly.

Let V be a finite-dimensional vector space over F, and let ip : V x V — F be an

F-valued F-bilinear symmetric form. We put (^[x] = (^(x, x). In this book we make

the convention that whenever we speak of ip of this type, the characteristic of F is

not 2. We call ip isotropic (on V) if p[x] = 0 for some x G V, ^ 0; we call /?

anisotropic (on V) if £[#] = 0 only for x — 0. (This means that p is anisotropic

if V — {0}.) We call a subspace U of V totally (^-isotropic if ip(x, y) = 0 for

every x, y G f/. We also call £ nondegenerate on V if there is no nonzero element

x EV such that p(x, V) = 0. For a subspace X of V on which ip is nondegenerate,

we put

(1.1) X± = {y eV\p(y, x) =0 for every x G X}.

This is a subspace of V, and is naturally called the orthogonal complement of

X in V with respect to (p. We use this symbol X1- whenever V and p are clear

from the context.

Assuming tp to be nondegenerate, we define groups O^ and SO^ by

Cv

=

o*{V) = 0((^, F) = {a G GL(V, F) | p[xa] = (p[x) for every x G V},

5 0 ^ = SC^(V) - 50(y?, V) - {a G O^(V) | det(a) - l } .

These are called the orthogonal group of (p and the special orthogonal group

of ip respectively. Let X be a subspace of V and ijj the restriction of ip to X.

If ip is nondegenerate, the symbols O^(X) and SO^(X) are meaningful. In our

later treatment, however, we shall often write these 0P(X) and SO^{X) without

introducing the symbol ip.

Given ip on V as above and a symmetric form ip' on a vector space V over

F, we say that (V, p) is isomorphic to {V, //) if £[#] =

^7[a:/]

for every x € V

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http://dx.doi.org/10.1090/surv/109/01