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Unformatted text preview: Math 220C June 4, 2008 Clifford Algebras Let K be a field of characteristic different from 2, let V be a vector space over K , and let Q be a quadratic form on V , that is, a map Q : V K such that Q ( v ) = 2 Q ( v ) for all K , v V . There is an associated bilinear form B : V V K defined by B ( u,v ) = 1 2 ( Q ( u + v )- Q ( u )- Q ( v )) , which is linear in both arguments. The Clifford algebra Cliff( V,Q ) is the free K-algebra with 1 generated by V , subject to the relations v v = Q ( v ) 1 for all v V . These imply u v + v u = 2 B ( u,v ) 1 for all u,v V , from which it is clear that the bilinear form B measures the failure of Cliff( V,Q ) to be an anti-commutative algebra. There is a natural map i : V Cliff( V,Q ), and it is not hard to see that this map is injective. The Clifford algebra has the property that if : V A is any K-linear map from V to a K-algebra with 1 satisfying ( v ) 2 = Q ( v )1 A , then there is a unique...
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