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Transposed algebras

Published online by Cambridge University Press:  20 January 2009

I. M. H. Etherington
I. M. H. Etherington
Affiliation:
Mathematicl Institute, The University, Edinburgh, 1.
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A linear algebra of order n, in general non-commutative and non-associative, may be regarded as being determined by the “cubic matrix ”consisting of its n3 constants of multiplication, and conversely. This requires that the n basis elements (units) of the algebra should be specified, and should be given in a definite order. Then the various “transpositions” of the cubic matrix induce corresponding “transpositions” of the algebra, for which a notation is given in §2.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 7 , Issue 2 , April 1945 , pp. 104 - 121
Copyright
Copyright © Edinburgh Mathematical Society 1945

References

page 104 note 1 For the definition of this and other terms used, Dickson, L. E., “Linear algebras,” Cambridge Tract No. 16, Cambridge, 1930, may be consulted; or the same author's “Algebren und ihre Zahlentheorie,” Zürich, 1927.Google Scholar

page 105 note 1 I refer by initials to ray earlier papers: N.C. = “On non-associative combinations,” Proc. Roy. Soc. Edinburgh, 59 (1939), 153–162;Google Scholar C.T.A. = “Commutative train algebras of ranks 2 and 3,” Journ. London Math. Soc., 15 (1940), 136–149;Google Scholar N.A.M.I.C. = “Some non–associative algebras in which the multiplication of indices is commutative,” ibid, 16 (1941), 48–55.

page 105 note 2 The suffixes run from 1 to n. It is to be understood throughout that Greek letters with suffixes denote elements of the field F, and that Σ indicates summation with respect to repeated suffixes (here with respect to k).

page 107 note 1 (Added 30 June, 1944.) An example is the algebra W of order m 2 with multiplication table (where suffixes run from 1 to m):

.

Albert has recently introduced, for algebras having an involution J, an operation which in some cases coincides with transposition. See Albert, A. A., “Algebras derived by non-associative matrix multiplication,” Amer. Journ. Math., 66 (1944), 30–40.CrossRefGoogle Scholar If M is the total matric algebra of order m 2 with multiplication table

.

andJ is matrix transposition (eij J = eji), then the algebras (in Albert's notation) M ρ (J), Mk (J), M kρ (J) are respectively M′, M′ and the above algebra W.

page 107 note 2 Dickson, L. E., “Linear Algebras,” loc. cit., §15, Theorem 3. In case X con. tains a modulus, the initial factors x in (L) and (R) can be omitted and the constant terms interpreted as multiples of the modulus; but we suppose that they are in any case included, since even if X contains a modulus its transposes may not.Google Scholar

page 108 note 1 Albert, A. A., “Non-associative algebras, I. Fundamental concepts and isotopy,” Ann. Math., 43 (1942), 685–707; §11. My X, Y, θ φ, ψ correspond to Albert's .CrossRefGoogle Scholar

page 119 note 1 Hausmann, B. A. and Oystein Ore, , “Theory of quasi-groups,” Amer. Journ. of Math., 59 (1937), 983–1004.CrossRefGoogle Scholar

page 119 note 2 Bruck, R. H., “Some results in the theory of quasi-groups,” Trans. Amer. Math. Soc., 55 (1944), 19–52.CrossRefGoogle Scholar

page 120 note 3 Murdoch, D. C., “Quasi-groups which satisfy certain generalized associative laws,” Amer. Journ. of Math., 61 (1939), 509–522.CrossRefGoogle Scholar