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Characterization of orthomaps on the Cayley plane. (English) Zbl 1406.51001

Let \({\mathbb P}_ 2 ({\mathbb O})\) be the projective plane over the real octonions equipped with the standard orthogonality. Using the description of \({\mathbb P}_ 2 ({\mathbb O})\) by the exceptional Jordan algebra \(H_3({\mathbb O})\) of Hermitian \(3\times 3\)-matrices with octonion entries, the point space is given by \({\mathcal P} = \{ P \in H_3({\mathbb O}) \mid P^2 = P \text{ and } \operatorname{tr}P = 1\}\). Two points \(P, Q \in {\mathcal P}\) are orthogonal if and only if \(\operatorname{tr}(P \circ Q) = 0\). The authors show that a map \(\phi: {\mathcal P} \to {\mathcal P}\) which preserves orthogonality is automatically bijective and preserves collinearity and orthogonality in both directions.

MSC:

51A10 Homomorphism, automorphism and dualities in linear incidence geometry
51A35 Non-Desarguesian affine and projective planes
15A63 Quadratic and bilinear forms, inner products
51P05 Classical or axiomatic geometry and physics
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
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References:

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