By J. Aguade, R. Kane

ISBN-10: 3540187294

ISBN-13: 9783540187295

Textual content: English, French

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**Extra info for Algebraic Topology, Barcelona 1986**

**Example text**

En } of E such that for any n x= ei xi ∈ E, i=1 we have n q(x) = ai xi2 , i=1 or equivalently, B(ei , ej ) = δij ai (1 ≤ i, j ≤ n). By definition, (q) = (−1) n(n−1) 2 n ai (mod(K × )2 ) i=1 is the “discriminant” of q. The construction of a Clifford algebra associated with a quadratic regular space (E, q) is based on the fundamental idea of taking the square root of a quadratic form, more precisely of writing q(x) as the square of a linear form ϕ on E such that for any x ∈ E, q(x) = (ϕ(x))2 .

For any a ∈ C(E, q) we often write ν(a) = a ∗ or a; ν is often called the conjugation of C(E, q). 1 Theorem Let e = {e1 , . . , en } be an orthogonal basis relative to q. With the above notation, we have the following relations: (ei )2 = ai (1 ≤ i ≤ n), ei ej + ej ei = 0 (1 ≤ i, j ≤ n, i = j ). , the linear space over K generated by the (ei1 · · · eim ), m even), C − = {ei1 · · · eim (i1 < · · · < im ), m odd}K . Since the two-sided ideal I (q) is generated by “even” elements, the deﬁnition of C ± is independent of the basis.

We have E1 = E1 , E2 = E2 , E1 E2 = E2 E1 = 0, E1 + E2 = 1 and eN = α(E1 − E2 ). C = C + E1 ⊕ C + E2 , the two components are both simple algebras, isomorphic to + C , and π is the automorphism of C that interchanges the units E1 and E2 and leaves invariant the elements of C + , and then interchanges C1 = C + E1 and C2 = C + E2 . Classical example: Let us assume that K = R and that the signature of q is (r, s). 4 Proposition according as R C + (r, s) ∼ C H 0, ±1 r − s ≡ ±2 (mod 8) ±3, 4 n(n−1) τ = (−1) 2 e , the principal antiautomorphism τ is of the ﬁrst kind for Since eN N C(r, s) if and only if n ≡ 3 (mod 4) and for C + (r, s) if and only if n ≡ 2 (mod 4).

### Algebraic Topology, Barcelona 1986 by J. Aguade, R. Kane

by Robert

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