Page 146 - Textos de Matemática Vol. 39

136 ANA PAULA SANTANA AND HELENA ALBUQUERQUE
product is defined by x.y = F (x, y)xy, ∀x,y∈G, where F is any 2-cochain on G (see[3]).Inthiscasethecocycleφ:G×G×G→K∗ isgivenby
φ(x, y, z) = F (x,y)F (xy,z) , ∀x,y,z∈G. F (y,z)F (x,yz)
Examples of KF(Z2n) algebras are Cayley algebras, e.g. octonions, and Clifford algebras, e.g. quaternions or complex numbers (see [3], [4]). In the case of Cayley algebras, the cochains F have the form F (x, y) = (−1)f (x,y), for some Z2-valued function f on G × G. For example, the following algebras are of this type, for the group and function f as indicated:
(1) The ‘complex number’ algebra:
G=Z2, f(x,y)=xy, x,y∈Z2,
where we identify G as the additive group Z2 but also make use of its
product.
(2) The quaternion algebra:
G=Z2 ×Z2, f(x,y)=x1y1 +(x1 +x2)y2, where x = (x1,x2), y = (y1,y2) ∈ G.
(3) The octonion algebra:
G=Z2 ×Z2 ×Z2, f(x,y)=xiyj +y1x2x3 +x1y2x3 +x1x2y3,
i≤j
where x = (x1,x2,x3), y = (y1,y2,y3) ∈ G.
To describe Clifford algebras as deformed group algebras KF(Z2n), we consider a vector space V of dimension n over a field K endowed with a non- degenerate quadratic form q, and the associated Clifford algebra C(V,q). It is known that there is an orthogonal basis {e1, ..., en} of V with q(ei) = qi. Then the algebra C(V, q) can be identified with KF (Z2n), where F is the 2-cochain defined by
n F(x,y)=(−1) j<ixiyj qxiyi,
where x = (x1,···xn), y = (y1,···yn) ∈ Z2n (see [2]).
i i=1