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Algebras related to Penrose-Smith transformations. (English) Zbl 0871.17031
A quadratic transformation \(Q\): \(a\to aQ\) in a finite dimensional vector space \(V\) gives rise to a commutative nonassociative algebra with product \(ab= {1\over 2} \{(a+b)^2 -a^2- b^2\}\), by taking \(a^2= aQ\). The author studies algebras obtained in this way from a class of quadratic transformations, studied by R. Penrose and C. A. B. Smith [Math. Proc. Camb. Philos. Soc. 89, 89-105 (1981; Zbl 0453.14005)]. In train and Gonshor algebras the sequences of principal powers satisfy tractable recurrence relations. Bernstein algebras are characterized by a simple condition on the plenary sequence. The author examines in succession the Mandel-, the Todd-Lyness transformations and finally the quadratic involution \(Q\) in the projective plane.
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