Continuous Transformation Groups

CONTINUOUS TRANSFORMATION GROUPS Essentially different from the groups already mentioned are the transformation groups studied by Sophus Lie and his pupils.

The system of n equations T : xi' =f X2, • • • , xnl a2, • • • , ar), (i= I, 2, 3, • • • , n) with parameters • • • , a, determine a transformation of the variables • • • , x„ into the variables • • • , x„', provided the equations can be solved for • • • , x„. The transformation may be written briefly T : xi' = f i(x (a), (i= 2, . • • , n) .

If, as the parameters a2, • • • , at vary, a+aa) - approaches zero with aa, the totality of all transformations thus obtained form a continuous transformation group of co' elements under the following conditions: (I) The functions fi considered as functions of • • • , x,, must be independent.

(2) The parameters

• • • , a,. must be essential, that is to say, a transformation cannot be expressed by means of a smaller number of parameters. For this it is necessary and suffi cient that the f's shall not all satisfy a differential equation of the form r of i(ai, a2, .. • , a,) = o.

i-1 aai (3) The product of two transformations T. and is expressible in the same form xi"=fi(xlc), (i= 2, • . . , n), where is a function of • • • , and • • • , alone for k= I, 2, • • • , r.

(4) The associative law for multiplication holds.

(5) The system of transformations contains the identical trans formation. That is, there is a system of values a?, • • • , ar of the parameters a such that From (5) it follows directly that to every transformation T of the group there is in the group an inverse transformation such that =I.

Infinitesimal Transformations.-Corresponding to every transformation group with r parameters, there is a general infinitesimal transformation also containing r parameters and giving rise to aDr-1 infinitesimal transformations. The group is generated by any r independent infinitesimal transformations. The determination of the infinitesimal transformations and their relation to the finite equations of the group are best shown by considering a particular group.

The group whose transformations have the form is called the symbol of the infinitesimal transformation. It is made up of the sum of the eight simpler ones, viz., p, q, xp, yp, xq, yq, each multiplied by the appropriate constant chosen from the set a, b, c, • • • . The finite equations for the group will be obtained by integrating the simultaneous system dx dy = dt ' subject to the initial condition that when t = o, x and yi = y. For two infinitesimal transformations, and it is easy to show that — U2(U1f) = (U12- U2 1)p+(U1r12- U2q1)q, which is again an infinitesimal transformation. The left member of this equation (Lie's Klammerausdruck) is denoted by Denoting by U2f, • • • , of the simpler infinitesimal transformations given above, one obtains for the general pro jective group of the plane the important result: (Ui Uk) = E Ciks f (i, k= I, 2, 3, • • • , 8) .

This formula is a special case of a very general theorem which asserts that the necessary and sufficient condition that r-inde pendent infinitesimals shall generate a group is that for every pair Uk the expression (Ui Uk) is expressible linearly in U2, • • • , The general theory which proceeds upon similar lines may be found in Lie's Continuierliche Gruppen or in ad mirably brief and succinct form in the article "Kontinuierliche Transformations Gruppen" in the Encyclopaedia der Mathemat ischcn Wissenschaften, by L. Maurer and H. Burkhardt.

Applications.

The applications of the theory of continuous transformation groups are manifold. One of the most important is the application to the theory of differential equations. A simple, but characteristic result, is that for every ordinary differential equation Xdx — Ydy = o, whose integral curves are transformed into themselves by an infinitesimal transformation = p+nq the expression is an integrating factor.

Lie made the theory the starting point for a profound study of the foundations of geometry. He reached the important result which may be stated as follows: Every real continuous group of transformations in a space of three or more dimensions whose points have free infinitesimal mobility may be transformed by real point transformations into a transitive n (n+ 1) -parameter 2 projective group of one of the three following kinds: (I) The group of Euclidean motions in the space.

(2) A projective group which leaves invariant the imaginary quadric surface x12+x22+ . . . I = 0.

(3) A projective group which leaves invariant the real quadric surface . . . - I = 0.

The first relates to the Euclidean or parabolic geometry, the second to the elliptic and the third to the hyperbolic geometry corresponding to spaces of constant zero, positive and negative curvature respectively. Each geometry is completely charac terized by its group.

In 1884 Poincare pointed out that to every complex number system taken over the field of ordinary complex numbers and having n linearly independent units, there corresponds a simply transitive group of homogeneous transformations linear in n parameters, and conversely. It follows that the theory of hyper complex numbers in the ordinary complex number field may be identified with the theory of certain types of continuous trans formation groups. (See L. E. Dickson, Algebras and their Arith metics Chap. III).

tions et des equations algebriques (187o) ; F. Klein, Vorlesungen fiber Ikosaeder and die Aufiosung der Gleichung von fun/ten Grade (Leipzig, 1894, trans. into English by G. C. Morris, 2d ed., 1913) E. Netto, Substitutionen Theorie and ihre Anwendung auf die Algebra (Leipzig, 1882, B. G. Teubner. Pp. 8-29o, trans. into English by F. N. Cole, 1892) ; H. Weber, Lehrbuch der Algebra (2d ed., Braunschweig, 1898-99) ; L. Bianchi, Lezioni sulla teoria dei gruppi di sostituzioni e delle equazioni algebriche secondo Galois (Pisa, 1900) ; H. Hilton, An Introduction to the Theory of Groups of Finite Order (1908) ; L. E. Dickson, Linear Groups with an Exposi tion of the Galois Field Theory (Leipzig, 1901) ; W. Burnside, Theory of Groups of Finite Order (2d ed., 1911) ; G. A. Miller, H. Blichfeldt and L. E. Dickson, Finite Groups (1916) ; H. F. Blichfeldt, Finite Collineation Groups (Chicago, 1917) ; A. Speiser, Die Theorie der Gruppen von endlichen Ordnung, mit Aufwendungen auf algebraischen Zahlen and Gleichungen so wie auf die Krystallographie (2d ed., Berlin, 1927) .

Infinite discrete groups: F. Klein and R. Fricke, Die Theorie der elliptische Modul Functionen (Leipzig, 189o-92, 2 vols., vol 1, pp. 19-764 treats of group theory foundations, and is an excellent supplement to Klein's Vorlesungen caber das lkosaeder).

Groups of continuous transformations: S. Lie, Vorlesungen caber continuirliche Gruppen mit geometrischen and anderen Anwendung Bearbeitet and herausgeben von Dr. Georg Schaffers (Leipzig, 1892 ) and Theorie der transformationen Gruppen, unter mit wirkung von Dr. Friedrich Engel, bearbeitet von Sophus Lie (Leipzig, 1892-93) ; J. E. Campbell, Introductory Treatise on Lie's Theory of Finite, Continuous Groups (1903) ; L. Bianchi, Lezioni sulla teoria dei Gruppi continui finite di transformation (Pisa, 1918). (E. B. S.)