clifford-algebras-II

clifford-algebras-II - Math 220C June 6 2008 Clifford...

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Unformatted text preview: Math 220C June 6, 2008 Clifford Algebras, Part II If Q is a nondegenerate quadratic form on a finite-dimensional C-vector space V , then in an appropriate basis B ( e i , e j ) = δ ij . If dim C V = n , then the corresponding Clifford algebra will be denoted by Cliff C n . Similarly, if Q is a nondegenerate quadratic form on a finite-dimensional R-vector space V , then in an appropriate basis B ( e i , e j ) = 0 for i 6 = j , while Q ( e 1 ) = ··· = Q ( e p ) = 1, and Q ( e p +1 ) = ··· = Q ( e p + q ) =- 1. In this case, the corresponding Clifford algebra will be denoted by Cliff ( p,q ) . Lemma. There are isomorphisms (1) C ( p,q ) ⊗ C (2 , 0) ∼ = C ( q +2 ,p ) (2) C ( p,q ) ⊗ C (0 , 2) ∼ = C ( q,p +2) (3) C ( p,q ) ⊗ C (1 , 1) ∼ = C ( p +1 ,q +1) Proof. (1) Let e 1 , . . . , e p + q be a basis for R ( p,q ) , f 1 , f 2 be a basis for R (2 , 0) , and e 1 , . . . , e p + q +2 be a basis for R ( q +2 ,p ) . Consider the map φ : R ( q +2 ,p ) → C ( p,q ) ⊗ C (2 ,...
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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.

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clifford-algebras-II - Math 220C June 6 2008 Clifford...

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