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On the Decomposition of an Infinitesimal Group

Published online by Cambridge University Press:  24 October 2008

J. H. C. Whitehead
J. H. C. Whitehead
Affiliation:
Balliol College, Oxford
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Extract

1. This note contains a second proof of a remarkable theorem, announced by W. Killing and first proved by Eugenio-Elia Levi, according to which any infinitesimal group G, in the sense of Lie, contains a subgroup which is simply isomorphic to the factor group G/Γ, where Γ is the greatest integrable invariant subgroup in G§. Also the theorem is sharpened in that it is proved for groups depending on real as well as on complex parameters. In the final section it is seen that the methods used in proving this theorem can be applied to the problem of solving certain equations which are stated in terms of any semi-simple infinitesimal group of linear operators.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 32 , Issue 2 , May 1936 , pp. 229 - 237
Copyright
Copyright © Cambridge Philosophical Society 1936

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References

† Math. Annalen, 36 (1890), 189.Google Scholar

‡ Atti Acc. Torino, 40 (1905), 3–17.Google Scholar Levi's proof depends upon Cartan's, E. classification of semi-simple groups (Thèse, Paris, 1894, 2nd ed. 1933). In his thesis Cartan proved the theorem in certain special cases (Chap. VI and p. 128, 2nd ed.).Google Scholar

§ See Cartan, , loc. cit. Chap. VI, theorem I.Google Scholar

║ Cf. Weyl, H., Math. Zeit. 23 (1925), 275.CrossRefGoogle Scholar The three parts of this paper, continued in Math. Zeit. 24 (1926), 328–95,CrossRefGoogle Scholar wi11 be referred to as Weyl I, II and III. For a general account of the structure of an infinitesimal group we Cartan, loc. cit., or Eisenhart, L. P., Continuous groups of transformations, Princeton (1933), Chap. IV.Google Scholar

Roman indices take the values 1, …, n and Greek indices, unless otherwise stated, the values, I, …, r.

† Indices will be raised and lowered in the ordinary way by means of the tensor gαβ, whose determinant does not vanish since the group g is semi-simple—Cartan, loc. cit. Chap. IV, theorem 1. In the absence of dots it is to be understood that indices are raised from the first lower to the last upper index and vice versa.

† Eisenhart, , loc. cit. p. 162.Google Scholar

‡ Since implies υσ = 0 the equations (1.2) have at most one set of solutions.

§ Cartan, , loc. cit. (2nd ed.), p. 113.Google Scholar

║ Notice that , regarded as a set of r vectors in a space of ½–r (r−1) dimensions, constitute a complete set of (independent) solutions of (1·5b), regarded as linear equations in the variables A λμ.

† Weyl, II, p. 375.Google Scholar

† Weyl, III, p. 381 and II, p. 289.Google Scholar See also Cartan, , La théorie des groupes finis et continus et l'analysas situs, Paris (1930), pp. 37–41.Google Scholar

‡ By a real subspace is meant one which contains the vector if it contains x It is not difficult to show that such a subspace is spanned by a set of vectors with real co-ordinates.

† Cartan, Thèse, Chap.IV, theorem 4.

‡ In other words, E/E1 is obtained from E by identifying any two vectors whose difference lies in E1.

† Weyl, III, pp. 360–1 and theorem 2, p. 364.Google Scholar The argument on p. 278 of Weyl I shows that Λ+α is a weight for at least one root a if Λ is a given weight.

† An equi-affine space of paths is one in which where ρ is a scalar density which may be taken to define volume. A necessary and sufficient for this to be the case is Bjk = Bkj where (see Eisenhart, L. P., Non-Riemannian geometry, New York (1929), p. 9).Google Scholar

‡ Ibid. p. 126.

† Cf. Hopf, H. and Rinow, W., Comment. Math. Heiv. 3 (1931), 209–25.CrossRefGoogle Scholar

‡ It is now to be understood that all the entities to which we refer are analytic and defined all over Vn. In particular each individual vector ξλ is defined at each point, and the operators (h1,…, hr) generate a finite group in Vn.

§ Actually hλf is a normal function whether f is normal or not.

║ See a forthcoming paper by Hodge, W. V. D., Journal London Math. Soc. 11 (1936).Google Scholar