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Duplication of linear algebras

Published online by Cambridge University Press:  20 January 2009

I. M. H. Etherington
I. M. H. Etherington
Affiliation:
Mathematical Institute, 16 Chambers Street, Edinburgh.
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The process of duplication of a linear algebra was defined in an earlier paper, where its occurrence in the symbolism of genetics was pointed out. The definition will now be repeated with an amplification. Although for purpose of illustration it is applied to the algebra of complex numbers, duplication will seem of no special significance if attention is fixed on algebras with associative multiplication and unique division; for duplication generally destroys these properties. The results to be proved, however, show that it is significant in connection with various other conceptions which appeared in the discussion of genetic algebras; namely baric algebras and train algebras (defined in G.A.), also nilpotent algebras, linear transformation and direct multiplication of algebras.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 6 , Issue 4 , October 1941 , pp. 222 - 230
Copyright
Copyright © Edinburgh Mathematical Society 1941

References

page 222 note 1 Etherington, , “Genetic algebras,” Proc. Roy. Soc., Edin., 59 (1939), 242–258.CrossRefGoogle Scholar Reference will also be made to “On non-associative combinations,” ibid., 153–162. These papers will be referred to as G.A. and N.C. Cf. also ibid. (B), 61 (1941), 24–42.

page 225 note 1 In the genetical symbolism, this theorem corresponds to the fact that in order to obtain the distribution of zygotic types of an rth filial generation, provided that no selection acts on the zygotes, it is sufficient to trace only the gametic distribution through the r–1 intervening generations.

page 226 note 1 van der Waerden, , Moderne Algebra (Berlin, 1930), I, pp. 56–57CrossRefGoogle Scholar, where, since the postulate of associative multiplication in rings is not used, the results apply to non-associative algebras. “Invariant subalgebra” is here called Ideal, and “difference algebra” Restklassenring.

page 228 note 1 See, e.g., Aitken, , Proc. London Math. Soc. (2), 38 (1935), 354–376.CrossRefGoogle Scholar

page 228 note 2 In this section, as in N.C. §3, δ, a denote positive integers.

page 228 note 3 Index is the usual word in this context: Cf. Wedderburn, , Proc. London Math. Soc. (2), 6 (1908), 77–118; p. 111.CrossRefGoogle Scholar But having drawn a distinction in N.C. between index and degree, I find the latter word more appropriate here. It is perhaps not irrelevant to point out an error in Wedderburn's paper, concerning nilpotent non--associative algebras. It is stated (loc. cit., p. 111) that the sum of all the rth powers of such an algebra is less than (i.e. is contained in but is not equal to) the sum of the (r − l)th powers. This is not true of the commutative algebra X = (a, b, c) where a 2 = b, ab = b 2 = c, ac = bc = c 2 = 0; for which X 2 = (b, c), X 3 = (c), X 4 = 0, X 2.2 = (c), X 5 = X 2.2+1 = X 3+2 = 0. For X is nilpotent of degree 5, whereas X 4 + X 2.2 = X 3. Cf. Etherington, , “Special train algebras,” Quart. Journ. Math, (in press).Google Scholar

page 230 note 1 The statement (G.A., p. 247) that “all the fundamental genetic algebras are special train algebras” refers to gametic algebras, not to the zygotic algebras which are derived from them by duplication.Google Scholar