By P. K. Jain, Ahmed Khalid

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Spin structures in covering spaces 43 on Q* and q : X --+ X* denotes the projection, then Q is the bundle q*(Q*) induced from Q* by means of this projection. Therefore, q*(P*) is a spin structure on Q on which F acts as a group of automorphisms. But, since irl(X) = 1, the spin structure q*(P*) is equivalent to (P,A). Thus, in summary, we conclude: Proposition. The spin structures in the principal bundle Q* over X* are in one-to-one correspondence with all left actions e of r on P satisfying e(y) = ly'.

We define an case by case. First case. n = 8k, 8k + 1. We have On = C2 ® ... ®(a(9,3) Second case. n = 8k + 2, 8k + 3. (2k times). We have On = C2 ® ... ® C2 (4k+1 times), and we set an = a ®(Q ®a) ®... ®(0 (3 a) (2k times). Third case. n = 8k + 4,8k + 5. We have A. = C2 ® ... ® C2 (4k+2 times), and we set an = (a (& ,6) ®... ®(a ®0) (2k + 1 times). Fourth case n = 8k + 6, 8k + 7. We have On = C2 ® ... 0(,c3(9 a) (2k+1times). 3, the properties listed above are easily checked. 1. Clifford Algebras and Spin Representation 32 Summarizing, we arrive at the following table for the real or quaternionic structures in An: an quaternionic structures real structures n commutes with Clifford multiplication 6,7 mod 8 n 2, 3 mod 8 n anti-commutes with Clifford multiplication 0, 1 mod 8 n 4, 5 mod 8 We now ask whether, in the case of an even dimension n, the structures just constructed are compatible with the decomposition of Dirac spinors On into the sum of Weyl spinors On ®On .

0 (3 a) (2k times). Third case. n = 8k + 4,8k + 5. We have A. = C2 ® ... ® C2 (4k+2 times), and we set an = (a (& ,6) ®... ®(a ®0) (2k + 1 times). Fourth case n = 8k + 6, 8k + 7. We have On = C2 ® ... 0(,c3(9 a) (2k+1times). 3, the properties listed above are easily checked. 1. Clifford Algebras and Spin Representation 32 Summarizing, we arrive at the following table for the real or quaternionic structures in An: an quaternionic structures real structures n commutes with Clifford multiplication 6,7 mod 8 n 2, 3 mod 8 n anti-commutes with Clifford multiplication 0, 1 mod 8 n 4, 5 mod 8 We now ask whether, in the case of an even dimension n, the structures just constructed are compatible with the decomposition of Dirac spinors On into the sum of Weyl spinors On ®On .