Commutative Train Algebras* of Ranks 2 and 3

Abstract

At the ond of this paper I gave without proof a multiplication table (7.25) which I wrongly stated to be a canonical form for a commutative train algebra of rank 3 with train root λ ≠ ½. Although it certainly determines an algebra of this type, it is not sufficiently general. For the commutative multiplication table I2 = I, Iu = λp, Iq = ½q, up = q, u2 = p2 = q2 = uq = pq = 0 is not included in (7.25); but it defines a commutative train algebra of rank 3, the rank equation of a normalised element X = I + αu + rβp + γq being X(X−l)(X−λ) = 0. The operational train equation is (if λ ≠ ½) X{Y−l}{Y−½}2{Y−λ} = 0; compare (7.10), contrast (7.26) loc. cit. As I did not make any use of (7.25), the results proved in the paper are unaffected. The above algebra is a special train algebra, in the canonical form which I have established for such algebrasf. In an earlier paper‡ I stated that a commutative train algebra of rank 3 is necessarily a special train algebra. This statement, which would have been an immediate corollary if the generality of (7.25) could have been established as I thought, may or may not be true.