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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 ( λv ) = λ 2 Q ( v ) for all λ ∈ 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 Kalgebra 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 anticommutative 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 φ : V → A is any Klinear map from V to a Kalgebra with 1 satisfying φ ( v ) 2 = Q ( v )1 A , then there is a unique...
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This note was uploaded on 08/06/2008 for the course MATH 220 taught by Professor Morrison during the Spring '08 term at UCSB.
 Spring '08
 MORRISON
 Algebra, Vector Space

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