Algebras related to Penrose-Smith transformations. (English) Zbl 0871.17031

Holgate, Philip

Nova J. Algebra Geom. 1, No. 2, 133-150 (1992).

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.

Reviewer: A.H.Boers (Rijswijk)

MSC:

17D92	Genetic algebras
51N10	Affine analytic geometry
92D10	Genetics and epigenetics
51N15	Projective analytic geometry

Keywords:

genetic algebras; Mandel transformations; quadratic transformation; commutative nonassociative algebra; Todd-Lyness transformations; quadratic involution; projective plane