# Clifford algebras, Clifford groups, and generalization of by Gallier J.

By Gallier J.

This path used to be learn within the division of computing device and knowledge technological know-how on the collage of Pennsylvania in 2002

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2. Springer, first edition, 1997. [11] Yvonne Choquet-Bruhat and C´ecile DeWitt-Morette. Analysis, Manifolds, and Physics, Part II: 92 Applications. North-Holland, first edition, 1989. [12] Morton L. Curtis. Matrix Groups. Universitext. Springer Verlag, second edition, 1984. [13] Jean Dieudonn´e. Sur les Groupes Classiques. Hermann, third edition, 1967. [14] William Fulton. Algebraic Topology, A first course. GTM No. 153. Springer Verlag, first edition, 1995. 43 44 BIBLIOGRAPHY [15] William Fulton and Joe Harris.

Observe that for 1 ≤ i, j ≤ n, we have f (ei )f (ej ) + f (ej )f (ei ) = (ei ej + ej ei ) ⊗ (e1 ee )2 = −2δij 1 ⊗ 1, since e1 e2 = −e2 e1 , (e1 )2 = −1, and (e2 )2 = −1, and ei ej = −ej ei , for all i = j, and (ei )2 = 1, for all i with 1 ≤ i ≤ n. Also, for n + 1 ≤ i, j ≤ n + 2, we have f (ei )f (ej ) + f (ej )f (ei ) = 1 ⊗ (ei−n ej−n + ej−n ei−n ) = −2δij 1 ⊗ 1, and f (ei )f (ek ) + f (ek )f (ei ) = 2ei ⊗ (e1 e2 en−k + en−k e1 e2 ) = 0, for 1 ≤ i, j ≤ n and n + 1 ≤ k ≤ n + 2 (since en−k = e1 or en−k = e2 ).

We now have the main periodicity theorem. 17 (Cartan/Bott) For all n ≥ 0, we have the following isomorphisms: Cl0,n+8 ≈ Cl0,n ⊗ Cl0,8 Cln+8,0 ≈ Cln,0 ⊗ Cl8,0 . Furthermore, Cl0,8 = Cl8,0 = R(16). Proof . 15 we have the isomorphisms Cl0,n+2 ≈ Cln,0 ⊗ Cl0,2 Cln+2,0 ≈ Cl0,n ⊗ Cl2,0 , and thus, Cl0,n+8 ≈ Cln+6,0 ⊗ Cl0,2 ≈ Cl0,n+4 ⊗ Cl2,0 ⊗ Cl0,2 ≈ · · · ≈ Cl0,n ⊗ Cl2,0 ⊗ Cl0,2 ⊗ Cl2,0 ⊗ Cl0,2 . 16, we get Cl2,0 ⊗ Cl0,2 ⊗ Cl2,0 ⊗ Cl0,2 ≈ H ⊗ H ⊗ R(2) ⊗ R(2) ≈ R(4) ⊗ R(4) ≈ R(16). The second isomorphism is proved in a similar fashion.