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clifford-algebras

clifford-algebras - Math 220C Clifford Algebras June 4 2008...

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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 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 φ : V A is any K -linear map from V to a K -algebra with 1 satisfying φ ( v ) 2 = Q ( v )1 A , then there is a unique homomorphism ˜ φ : Cliff( V, Q ) A such that ˜ φ i = φ . In fact, Cliff( V, Q ) can be
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