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A* is defined by the power series of the exponential function: °O exp(a) a an - 1: mt' n=O We are going to apply this observation in the case of the real Clifford algebra Cn.

In On there exists a quaternionic Spin(n)equivariant structure an : On - On which anti-commutes with Clifford multiplication: an(x XEll Bn and 0EOn. 4) Let n = 8k + 6, 8k + 7. In On there exists a real Spin(n)-equivariant structure an : On -> On which commutes with Clifford multiplication: xEl8n and 2b EOn. Proof. 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) ®...

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