hesses's principle of transfer and the representation of lie algebras

Hesses's Principle of Transfer and the Representation of Lie Algebras

THOMAS H A W K I N S

Communicated by B. L. VAN DER WAERDEN

A well known feature of the representation theory of a finite dimensional semisimple LIE algebra fl over an algebraically closed field of zero characteristic is that although there exist an infinite number of finite dimensional irreducible .q modules V(~)--one for each dominant weight ~ - - they can all be generated from the I irreducible fl modules V(~i) corresponding to the fundamental domi-

nant weights ~i. That is, if ~ ~ m~xs, V(~) is the submodule of the tensor product ~ l

V(~l) ® ... ® V(~0 ® ... ® V(~3 ® ... ® V(~3

m ~ m I

generated by applying all the elements of the universal enveloping algebra of .q to

t,~ (x) . . . @ r~ ® . . . :x~ z't (x; . . . @ v~

m z I~t l

where v,-C V(~) is a highest weight vector. The module V(~) defined in this manner is sometimes referred to as a CARTAN product of the modules V(~), and the appelation is appropriate since the above construction had its origins in a seminal paper [1913] by I~LIE CARTAN. The CARTAN product construction was vital to CARTAN'S proof that every dominant weight associated to a semisimple LIE algebra determines an irreducible representation, but he did not conceive of this construction as we do today. Tensor algebra as such was nonexistent in 1913. It was not until after WEYL'S mathematical exposition of the theory of general relativity in the many editions of Raum Zeit Materie that began appearing in 1918 that tensor algebra gradually became a part of basic mathematics. From CARTA~'S own conceptual point of view in 1913, his construction was based upon a generalization of a geometrical Principle of Transfer (Uebertragungsprinzip) which OTTO HESSE had introduced in 1866.

42 T. HAWKINS

The purpose of this paper is to unravel the historical thread connecting HESSE'S original Principle with the version conceived by CARTAN -- a version that HESSE would never have recognized since it lived in a conceptual world far removed from his own. The historical unravelling has several side benefits. The fate of HESSE'S Principle provides an instructive case study of the metamorphosis of a mathematical idea while at the same time threading together several episodes in the history of the representation of LIE algebras. HESSE'S Principle functions in this essay as a "historical mirror" in which to view certain aspects of the history of the representation theory of LIE algebras in a way that conveys the flavor of the nineteenth century mathematics that nurtured its development.1 In particular, it serves to evoke a part of the geometrical framework within which CARTAN created his important trilogy of memoirs [1913, 1914a, 1914b] on what would now be described as the representation theory of real and complex semisimple LIE algebras. By the time he composed these memoirs he had become deeply involved with geometrical applications of LIE'S theory of groups. One manifesta- tion of this involvement was his acceptance of the tasks of revising G. FANO'S Enzyklop~idie article [1907] on the application of continuous groups to the classification of geometries in the sense of KLEIN'S Erlanger Programm. The task brought CARTAN into contact with an extensive corpus of geometrically related research that was of potential relevance to LIE'S theory. In composing his trilogy of memoirs, he drew upon it for inspiration, for technical ideas to prove his theorems and for geometrical interpretations of his results. His treatment of HEssE's Prin- ciple is a case in point. Another side benefit of the essay is that it sheds some new light on the influence of the Erlanger Programm. 2 As will be seen, the general perspective of the Erlanger Programm, especially its treatment of HESSE'S Principle, figured prominently in its metamorphosis.

Unless otherwise noted, all groups, LIE algebras and projective spaces are considered with respect to the field C of complex numbers. For example the notation P" will be used for complex n-dimensional projective space, and PGL(n + 1) will denote the general linear group of projective transformations of pn. In the nineteenth century this group was conceived of as consisting of all transformations of the form

, ai, l x I + . . . + a i , x , 4- ai,,+l X i ~ " , i ~ 1 , . . . , n ,

a ,+l , l x l 4- . . . 4- a ,+ l , , x , 4- a ,+l , ,+i

such that laijI @ o. In homogeneous coordinates Xi, i ---- 1, . . . , n 4- 1, where xi = X i / X , + I , the transformations were expressed in the form

~X[ = ai, l X 1 4- . . . 4- ai,n+lXn+l, i = 1 . . . . . n 4- 1.

Notations such as fl, r, ~l(n 4- l), gl(n 4- 1), etc. will be used to denote LIE algebras over C.

1 Regarding historical mirrors see also HAWKINS [1987]. 2 Regarding the historical problems relating to the influence of the Erlanger Programm

see HAWKINS [1984].

Hesse's Principle of Transfer 43

LUDWIG OTTO HESSE (1811-1874) received his mathematical education and orientation in the mathematical school that grew up around JACOBI at the Uni- versity of K6nigsberg. From JACOBI HESSE learned the elegant algebraic analysis exemplified by JACOBfS formulation of the theory of determinants. The goal of HESSE'S work became that of applying the new algebraic analysis to geometry, to study what he called analytic geometry but would now be classified as algebraic geometry. "1 have set for myself the problem," HESSE wrote to his former student ARONHOLD, " o f exploiting the treasures of Geometry on behalf of Analysis. ''3 The study of geometry by means of elegant algebraic analysis was pursued by HESSE on two fronts, which gradually merged: through research papers and through his lectures on geometry, which he began publishing in 1861. According to another former student of HESSE'S, A. CLEBSCH, these lectures were the first"signs of life'" exhibited by "modern algebra" outside the narrow confines of mathematical journals. 4 By the time HESSE became a Full Professor at the University of Heidel- berg in 1856, his research was largely a byproduct of his efforts to develop and perfect his lectures. His work on the Principle of Transfer typified his research program during the Heidelberg years.

HESSE introduced his Principle of Transfer in three publications all appearing in 1866. It was presented in the third and fourth lectures of a paper in Zeitschr[ft

far Mathematik und Physik entitled "Vier Vorlesungen aus der analytischen Geo- metrie" which was also published separately as a monograph [1866a]. The contents of the fourth lecture, which presented an analytical reformulation of the Principle, he deemed noteworthy enough to publish in CRELLE'S Journal [1866 bl.S To HESSE his Principle was an analog of the well-known Principle of Duality for the projective geometry of the plane, which he also referred to as a principle of transfer [1866a: 32]. Consider for example

Pascal's Theorem. I f a simple hexagon 6 & inscribed in a nondegenerate conic, ttle points of intersection of its opposite sides are collinear.

By the Principle of Duality, PASCAL'S Theorem transfers into its dual, which in this case is

Brianchon's Theorem. If a simple hexagon has elements of a nondegenerate line conic 7 for its sides, then the lines joining opposite vertices are concurrent.

Thus for every theorem proved, a second is obtained by transference; and for this reason HESSE deemed the discovery of principles of transfer as far more geometrically important than individual theorems [1866a: 32].

a Translated from a letter dated 1 November 1849 [HESSE 1897: 717]. '~ Translated from a letter dated 24 February 1862 [HESSE 1897: 719]. s Apparently the Principle was known to Hesse as early as 1851 when he communi-

cated it to JACOB1 in letter [HEssE 1897: 706]. 6 A hexagon is simple if no three consecutive vertices are collinear.

A line conic is the family of tangent lines to a point conic. Hence the hexagon cir- cumscribes this point conic.

44 T. HAWKINS

HESSE was apparently led to his own Principle of Transfer by perceiving an analogy between the theorems of PASCAL and BRIANCHON, o n the one hand, and two theorems on involutions of the projective line which he published in [1864]. Before stating these theorems it will be helpful to explain some terminology. Ex- pressed in modern terms (which HESSE did not employ) an involution is defined by a TC PGL(2) such that T 2 ~ 1. Three pairs of points on line (i.e., points of p1) are said to be in involution if an involution T exists such that T takes each point of each pair into the other point of the pair. Given a pair of points o¢, fl a pair of points 7, 6 are said to be harmonic conjugates with respect to ~, fl if the cross ratio R(o~, fl, 7, 6) ~ -- 1. We shall express this fact with the notation H(c~, fl; 7, 6). HESSE'S theorems [1864: 519, 521] may be stated as follows:

Theorem A. Given any six points 21 . . . . . 26 on a line, there exists a pair o f points (~, fl) which is simultaneously in involution with the following pairs: (i) {21, 22), {2,, 25); (ii) (22, 23) , 0.5, 26) ; (iii) (2a, 2,), (26, 21).

Theorem B. Given any six po&ts 2~ . . . . . . 26 on a line, construct the following point pairs: (i) {~, fl) such that H(o~, fl;2L, 22) and H(o~, fl;24,25); 0i) {y, 6) such that H(7, 6; 22, 23) and H(y, 6; 24, 26) ; (iii) (t z, v) such that H(/z, v; 23, 24) and H(#, v; 26, 21). Then the three point pairs (o¢, fl), 47, 6), {~t, v) are in involution.

With the aid of algebraic analysis, HESSE discovered that these theorems are "duals" of the theorems of PASCAL and BRIANCHON with respect to a new Principle of Transfer, which we present in the second, more algebraic, of his two formulations. It was this formulation that he also published in CRELLE'S journal [1866b].

HESSE'S idea was to set up a one-to-one correspondence between points P (x, y) in the plane and pairs of points p ~ ~2~, 22) belonging to the line, which he referred to as the fundamental line [1866 a: 49 ft. ; 1966 b: 531 ft.]. The correspondence, which we shall denote by q~: P - + p , was established in the following manner. Let A, B and C denote linear functions of the Cartesian coordinates x, y of P,

(1) A ~- ax q- a'y -k a", B - bx -k b'y q- b", C = cx ~- cy' -k c",

and consider the quadratic equation

(2) 4~(2; x, y) = A2 z q- B2 -k C ~- O.

Then each fixed point P--~ (x, y) can be made to correspond to the pair p ---- {2~, 22) consisting of the roots of (2): "To every point of the plane there corresponds a point pair on the fundamental line and conversely to every point pair on the fundamental line there corresponds a point of the plane" [1866b: 532]. This is correct if "plane" and "fundamental line" are interpreted as the complex projective plane and line, respectively, as HESSE intended, and if the matrix of coefficients in (1) is nonsingular. 8

8 Probably HESSE conceived of these coefficients as having "general values" as was common in the "generic reasoning" that prevailed at this time. On generic reasoning in algebra, see HAW~:INS [1977a, 1977b.]


Having thus set up the correspondence q~ : P -7 p, HESSE began by considering how configurations and relations in the plane "transfer" under ~. In this regard the locus of all points P(x, y) in the plane for which the discriminant of the quadratic (2) vanishes played a prominent role. These points are defined by the equation

(3) B 2 -- 4AC =- O,

which, as an equation in x and y, defines a conic section ~. HESSE called ~ the directrix. Since the points P on ~ are precisely those for which the quadratic (2) has a double root, it follows that q~[~] consists of all "double point pairs'" p -- (2, 2}.

Using results he had derived on involutions [1864: 179-85], HESSE proved that P, P', P " are collinear if and only if p, p', p" are in involution [1866b: 533]. Stated in more familiar terms and with the necessary qualifications, HESSE'S result is that if 5 ° is any line that is not tangent to the directrix ~ , then an involution T~ PGL(2) exists so that q~[L~'] consists of all p - {2~, 22} such that T(21) = 22. If P, P' are the (possibly imaginary) points of intersection of 5O and ~ , then p -- ~ (P ) and p' = q)(P') are double point pairs and so of the form p -= {2, 2}, p' = [2', 2'}, and ,t, ,1' are the fixed points of T. From known properties of involutions, it followed that if T(#) -- v, then H(2, 2';/~, v). Hence HESSE could characterize q~[so] as consisting of all pairs {/~, ~,} such that H(2, `1';/~, v). For a line 5O which is tangent to @ at Po, HESSE showed that q~[so] consists of all pairs {20, 2}, where {20, )to} -: q) l(Po) and ,t ~ pl .

By the Principle of Transfer embodied in the correspondence q~, the theorems o f PASCAL and BRIANCHON transfer to HESSE'S Theorems A and B. Consider PASCAL'S Theorem, which is transferred into Theorem B as follows. 9 Let the conic in PASCAL'S Theorem be the directrix ~ , and let P~ . . . . . P6 be the vertices of the hexagon. (It is helpful to draw a picture.) Since the Pi ~ ~ , we may write ~b(Pi) ~ {2~, ̀ 1i}. Let P, Q, R be the points of intersection of the line pairs PIP2 and P4P s, P2Pa and PsP6, P3P,~ and P6P1, and set qS(p) = {x, [;t}, ~(Q) [7, d~, q)(R) = [/~, ~}. (Then {~,/3}C q)[PtP2] implies, by HESSE'S characterization of q)[so] for 5O not tangent to ~ , that H(`1~, 22 ; :x, fl), which is (i) of Theorem B. Parts (ii) and (iii) follow in similar fashion. Finally, since P, Q, R are collinear. if 5 ° is the line they determine (usually called the PASCAL line) HESSE'S characterization of q515o] implies the existence of an involution T such that the conclusion of Theorem B holds. Now consider BRIANCHON'S Theorem. Let Pj . . . . . P(, denote the verticies of the hexagon, and let ~ be the point conic whose tangents form the line conic of BRIANCHON'S Theorem. Then if Qi~ ~ is the point of tangency of PiPi+ ~ (where Pv Pl), we may write ~ ( Q i ) = {2i, ,1i}. From HESSE'S characterization of q~[so] when 5O is tangent to @, it follows that ~(P~) [,1L, ,12}, ¢/)(P2) = (,12, `13} . . . . . t])(P6) : [,16, ,1~}- Let R denote the point in which by BRIANCHON'S Theorem the lines P~P4, P2Ps and P3P6 meet. If r - (1)(R) [a, fl}, then it follows that since, for example, rE q)[P~P4], the pair r ~ , t:~'~ is in involution with the pairs [,1~, ,12} and {,1,, ,is}. And so on.

o Using his first formulation of his Principle of Transfer, HESSE transferred PASCAL'S Theorem into Theorem A [1866a: 40-41].

46 T. HAWKINS

HESSE concluded his discussion of his Principle by claiming that it would make possible the discovery of many new theorems about the projective geometry of the straight line. Once such theorems were discovered it would be important, he felt, to supply them with direct and natural proofs. Then " the broader basis of the plane can be left behind. One can limit oneself to the line without surrendering lordship of the plane" [1866 a: 57; 1866 b : 538]. No one, including HESSE himself, seems to have attempted to carry out HESSE'S envisioned program for the projective geometry of the line. lO But that does not mean that HESSE'S Principle was not well known among nineteenth century mathematicians. HESSE'S writings, especially his lectures, were standard reading for mathematicians interested in the relations between algebra and geometry. In particular there are many references to HESSE'S Principle in the writings of nineteenth century mathematicians. Some mathematicians showed how HESSE'S Principle could be derived "most simply" or "most naturally" from their own, quite different mathematical viewpoint, some generalized and applied part or all of the ideas involved in the Principle and others introduced their own analogous principles of transfer. A study of all references to HESSE'S Principle would provide an interesting cross section of nineteenth century mathematics, but would detract from our main objective which is to unravel the historical thread linking HESSE'S original Principle to the generalizations of it given by CAR- TAN. Thus we shall restrict our attention to those treatments of HESSE'S Principle which are relevant to our objective. The first of these is found in FELIX KLEIN'S Erlanger Programm [1872], and it also appears to be the first treatment whatsoever of the Principle.

When KLEIN wrote the Erlanger Programm he was in close contact with SoPnus LIE, who shared many of the views set forth within its pages. Since LIE also contributed to the metamorphosis of HESSE'S Principle, it is helpful to begin with a few remarks about their relationship. KLEIN and LIE first met in 1869 at the University of Berlin. KLEIN had studied mathematics with PLOCKER in Bonn, and LIE, although he had never met PLi3CKER, had studied his books and consequently considered himself as one of his students. KLEIN and LIE thus shared a very intuitive, geometrical way of conceiving mathematics that stood in sharp contrast to the analytical and critical spirit that tended to pervade the Berlin school of WEIERSTRASS. Finding themselves intellectually isolated in Berlin, KLEIN and LIE headed for Paris where they discovered a greater intellectual affinity with mathematicians such as GASTON DARBOUX. They also met CAMILLE JORDAN, whose book, TraitO des substitutions [1870] appeared while they were in Paris. JORDAN'S book was the first systematic and comprehensive treatment of the theory of finite permutation groups and its applications, including above all applications to GALOIS' theory of equations but also applications to geometry. Although it is unclear to what extent they digested its contents, JORDAN'S book inspired them to call for the creation of an analogous theory of continuous groups, which they envisioned as having many applications in the study of differential equations and geometry. KLEIN'S Erlanger Programm was written with that vision in mind,

lo Apparently HESSE'S only further involvement with his Principle was to consider the nature of the set ~-115e], where 5e consists of all {o~,/3} such that c~ and /3 are the end- points of segments of constant length [HEssE 1897: 649-650].


and a few years later L1E began to create such a theory, encouraged by his discovery of the connection between a continuous group of transformations and the associated "g roup" (viz. LIE algebra) comprised by its infinitesimal transformations.

In the Erlanger Programm KLE1N put forward the idea that the diverse geometrical methods that had proliferated during the nineteenth century could be characterized by specifying, first, a "mani fo ld" of real or complex n-tuples which coordinatized some geometrical element, such as points or lines or spheres, and by specifying in addition a group of transformations which act upon the manifold and define the permissible "motions" . Thus for KLEIN, a geometrical mode of treatment, or a "geometry" as we shall say, was given by describing a manifold ;)2 and a group G of transformations acting on it. The geometrical properties of the geometry were defined to be those properties of configurations invariant under all transformations of G. In this way KLEIN sought to bring conceptual unity to the apparently disparate corpus of nineteenth century geometrical research. Hand in hand with his characterization of a geometry went his notion of the equivalence of geometries. A geometry defined by ;)L G was said to be equivalent to

the geometry ,~}~, G if a one to one, onto mapping (Abbildung) ~u : .~)I , :~ exists

such that () consists of all 7 ~ *PT~P ~ for some T~ G. KLEIN regarded the

geometries ;)I, G and ;75, G to be equivalent since every theorem of the former corresponds, through the intermediary of the mapping tp to a theorem in the latter and conversely.

KLEIN was familiar with HESSE'S Principle of Transfer and viewed it as an exemplification of his more general ideas. Indeed, the above notion of equivalence was described under the heading "Ueber t ragung durch Abbildung" ("Transfe- rence by Mapping"), and HESSE'S correspondence O defines a mapping tp in Kt.EI~'S

sense of the word, with ~1~ -- p2, G -- PGL(3) and ,~J} = O[:1~], G -- ~tJGtl j However KLEIN seized upon an observation made by HESSE in passing to stress a different way of interpreting HESSE'S ideas in terms of his general framework. HESSE had observed that the double point pairs p -- {2, 2] could be identified with the points 2 of the projective line P~ [1866a: 52; 1886b: 533]. This observation was made in passing to justify regarding the double point pairs as a fundamental class of point pairs. Of course, corresponding to this class in the plane is the directrix ~ . Thus implicit on HESSE'S remark is a mapping ~ : P~ > ~ , namely

(4) 7s(,~) __ O 1([/~, )~}).

The projective geometry of the line is thus seen to be equivalent in KLEIN'S sense

to the geometry on ;~ ~ determined by the group ( ~ - - 7 / P G L ( 2 ) g J i According to KLEIN, this equivalence "coincides in essence with the principle of transfer proposed by Hesse . . . " [1872: 471-472]. We shall refer to the principle implicit in (4) as the Hesse-Klein Principle o f Transfer. This version of HESSE'S Principle became the basis for much of its further generalization.

When he presented his Principle of Transfer in 1866, HESSE had pointed out that "in a similar manner the geometry of space can be reduced to that of the straight fine and even the geometry of more than three dimensions. In that case systems of three or more points are involved rather than points pairs" [1866a:

48 T. HAWKINS

50; 1866b: 531]. It is easy to see how to generalize the original Principle along the lines HESSE suggested. To obtain a correspondence q~: P - + p between P = (xl . . . . . xn) and p = (21 . . . . ,2,}, let Ai, i = O, 1 . . . . . n, be linear functions of the x i and let the 2~ be the roots of

(5) 9 ( 2 , X) ~- A o 2n ~- A L 2 n 1 ~_ . . . _~_ An = O.

As in the case n = 2, the actual correspondence ~ depends upon a fixed choice of the A i. A convenient choice is to let

(") A i = i Xi' i = O, 1 . . . . . n, x o = 1.

Thus (5) becomes

(6) q~ = Xi 2 n - i : O. i:0 i

The analog of HESSE'S mapping of the fundamental line onto a conic section is obtained by identifying the points of the line with the n-fold roots A~ = ... . . . . 2 n = t of (6). In this case

which, upon comparison with (6), shows that p = {t . . . . . t} corresponds to P = (xl . . . . . xn) , where

(7) x i : ( - - t ) i, i = 1 , . . . , n .

Thus the points on the fundamental line are made to correspond to the points on the curve c~ n defined by (7). This is the rational normal curve of order n in pn.ll In particular, c~2 : ~ , HESSE'S directrix, which in this case is the parabola xz : x~. I t should be noted that if homogeneous coordinates 2 : ut : u2 are introduced on the line and homogeneous coordinates xi = a i : a o in n-dimensional space, then (6) becomes

(6') q~ : i alUl bl 2 ~---- O, i : 0

and ~ is now a binary form whose coefficients a i represent the homogeneous coordinates of a point P in n-dimensional projective space. Thus HESSE'S Principle sug-

2 ( n ) n - i i with points P - gests the idea of identifying forms such as ~v : a~ul u 2 i=0 i

(ao . . . . . an) E pn. Some mathematicians regarded HESSE'S Principle as the source of this idea. For example, C. ST~PHANOS began his "M6moire sur la repr&entation des homographies binaires par des points de l 'espace . . . , " by writing: " I t has been some time now since the idea of representing diverse algebraic forms ... by points of the plane or of space assumed an important place in the domain of algebraico-

11 For the significance of this curve in algebraic geometry, see SEMPLE • ROTH [1949: 34.]


geometric considerations. Through his Uebertragungsprincip, HESSE was the first to teach the representation of quadratic binary forms by the points of the plane of a conic. Since then the projective study of groups of points on a rational curve ... has been linked to the representation of diverse binary forms ... by points of the plane or of space" [1883: 299].

The idea of representing forms as points, when extended to n-dimensional space, became an essential ingredient in the metamorphosis of HESSE'S Principle into the version given by CARTAN. It underlies the first treatment of the n-dimensional version of the Principle in W. F. MEYER'S book, Apolaritiit und rationale Kurven. Eine systematische Voruntersuehung zu einer allgemeinen Theorie der linearen R~iume [1883]. As the subtitle suggests, MEYER intended his book as providing the first steps towards the algebraic study of n-dimensional projective geometry. For MEYER and his contemporaries, the theory of binary forms and their invariants represented the most recent and powerful algebraic tool with which to explore n-dimensional projective geometry. However, the theory of binary forms was usually considered as the algebraic equivalent of the projective geometry of the line, p l That is, the binary form ~0 defined by (6') can be identified with the con- figuration in P~ consisting of its n roots as defined by (6) or (6'). Sets of n-point configurations left invariant by PGL(2) define projective properties. Equations l(a) = 0, where I(a) is an invariant of the form 9~ define such sets. This identification of the invariant theory of binary forms with the geometry of the projective line is taken for granted in KLEIN'S Erlanger Programm. HESSE'S Principle provided the means of extending the application of the theory of binary forms and their invariants to n dimensional projective geometry. That is, given a binary formq~ let p = {21 . . . . . 2,} be the n-tuple of its roots. Then we may associate with p (and hence with q~) any of the following: (i) the n points of P" on the normal curve ~n which are obtained by successively setting t = 2 k in (7) for k = 1 . . . . . n ; (i) the

hyperplane ~ ( n ) (-- 1) i a, iYi = 0 passing through the n points of (i); (iii) /

the i::0 . i

point (ao . . . . . a,), which is the "pole" of the hyperplane in (ii) with respect to normal curve c~,. Depending on the context, each of these modes of identification had its advantages. Thus HESSE'S Principle became associated with the theory of binary forms and the identification of such forms with points (and hyperplanes) in n-dimensional projective space. Although such an association is made in MEYER'S book, he himself tended to stress the role of the normal curve (rather than the form q~) in obtaining the above coordinatizations of the points and hyperplanes in P~ [1883: xiv, 42]. Such an association was given greater emphasis and further developed in the work of SEGRE [1885] and STUDY [1886, 1889] to be discussed below, but first we consider the manner in which KLEIN and LIE developed the ideas surrounding HESSE'S Principle in i1 dimensions. Although their ideas were first published in 1893, they were conceived earlier and should be kept in mind when the work of SEG~E and SI"UDY is considered.

Since the Erlanger Programm was addressed to the faculty at the University of Erlangen as part of the formalities of KLEIN'S appointment to a professorship there, mathematical details were not given. In fact, after its publication in 1872 KLEIN'S interests turned in other directions, and it was not until 1892-93 that he

50 T. HAWKINS

presented a detailed exposition of its ideas. This occurred in his lectures at G6ttin- gen [ 1893 a, 1893 b]. KLEIN'S renewed interest in the ideas of his Erlanger Programm was generated in part by the appearence of Lm's Theorie der Transformationsgrup- pen [1888, 1890, 1893]. These weighty tomes represented LIE's first attempt, with the help of his assistant F. ENGEL, at a comprehensive, systematic exposition of his theory of transformation groups and encouraged KLEIN to judge the time to be ripe for the further development of his ideas in conjunction with LIE'S. We now consider how each had developed the idea of the HESSE-KELlY Principle of Transfer of 1872. We begin with KLEIN.

In his lectures [1893a: 388-391] KLEIN considered the the n-dimensional analogue of the HESSE-KLEIN Principle of Transfer defined by the mapping ~ of (4). The n-dimensional analogue of ~ is a mapping 5u, : pl __> g~ which may be defined by ~ ( 2 ) -- (2, 22 . . . . . 2n), or, in homogeneous coordinates by

(8) ~(u~, u9 = (.~, .~' '"2 . . . . . ug .

Thus the projective geometry of the straight line is transferred to a geometry on

~ , defined by the group (7, = W,, : PGL(2) ,, W, i. As KLEIN showed in his lec-

tures, if TE PGL(2) is defined by ~u~-- x~uz +-~2u2, i = 1, 2, then 7~: (x0, . . . , x,,) t t -~ (% . . . . . x,), where xi ~-u~ iu~ and x~ = (u~)" i(u~)i so that the equations

for 7 ~ turn out to have the form o'x ' /~: ~ /4six j, i = O, I . . . . . n. In other words, j 0

(;, defines a subgroup of PGL(n i 1), which we shall denote by H~ in honor of HzSSE. H, is isomorphic to PGL(2) and has the property that it leaves the normal curve c6~, invariant: T[~,] ( g ' , for all TE H,,. KLEIN and LIE also realized this property characterizes H, in the sense that H,, consists precisely of all TE PGL(n + I) such that T[C~,] Q c6',,. LIE and his students frequently referred to H,, as the group of the normal curve in n dimensions, and since ~z is a conic, H2 was called the conic section group. KLEIN pointed out that this con- sequence of the Principle of Transfer- - the generation of the groups H, -- was "no tewor thy" because "these groups of collineations ~z are extraordinarily important in the theory of groups. Thus it follows that not only does a three term group ~3 of transformations of the plane exist which takes a conic section into itself, something we knew previously, but likewise in space a three-fold infinite group relative to a C3 and so on" [1893a: 390-391].

The "extraordinary importance" of the groups H, for LIE'S theory was due to further properties of these groups discovered by LIE and by EDUARD STUDY. LIE realized that H, has the geometrical property that it "leaves nothing planar invariant". That is, in acting upon P~ there is no point, line, plane, etc. which is taken into itself by all the TE H~. To convey the flavor of the methods LiE had introdced in creating his theory of groups, we briefly consider how he derived this property for //3 in his lectures on continuous groups [1893a].

In Cartesian coordinates a basis for the infinitesimal transformations (or LIE algebra) of H 3 consists of XI = p ~ 2xq i 3yr, X2 : xp 2yq + 3zr, X3 3(xZp -4- xyp + xzr) -- 2yp -- zq, where we have followed DE's convention of

~z That is, projective transformations. ~3 That is, a three dimensional group.


writing p = b/~x, q = ?/~), and r = ~/~z [1893a: 416]. In studying transformation groups with an eye towards applications, LIE concerned himself with the problem of determining the manifolds left invariant by a group. This geometrical problem was regarded as fundamental for applications to differential equations as well as to geometry [1893a: 404-5]. Suppose now that X £P ~ ~tq + ;-r is any infinitesimal transformation in three dimensions, where ~:, ~j and ~," are functions of x, y, z. Then the curve through some point P (x, y, z) satisfying the differential equations dx/dt = ~, dy/dt = ,r l, dz/dt = 2 is called the Bahnkurt, e of X through P. In effect it represents the orbit of P under the one-parameter group ot transformations generated by X. This curve has a tangent vector at P which is T(P) = ~i 4- ~]] + ~k. Consider now H3. The Bahnkurve o f X = el.X, ? e2Xz -~- e3X3 through P has tangent vector el T l ( P ) - + e z T z ( P ) - + e3T3(T) , where T~(P) denotes the tangent vector at P to the Bahnkurve determined by X~, i 1,2, 3. Thus the linear span of the Tg(P) yields all possible directions in which P can be moved by the elements in Ha. For this reason LIE considered the matrix with rows given by the components of the Ti(P):

M(P) = x 23' 3z .

3x 2 - 2 ) , 3xj . . . . z 3xz

At any point where the determinant of M ( P ) vanishes there are at most two degrees of freedom for a possible Bahnkurve through P. Computat ion of this determinant yields the equation of the surface of fourth order

3(4xZz -- 3xZy 2 @ 4l ,3 -- 6xyz ~ Z 2) -- O.

This surface contains all the points P such that, for any infinitesimal transformation X of H3, the tangent T(P) to the Bahnkurve lies in the tangent plane of the surface at P; thus the Bahnkurve lies on the surface. In other words this surface is taken into itself by Ha and it is the only two dimensional manifold with this property. To find one dimensional manifolds on this surface which are left invariant by Ha one seeks all the points P for which M(P) has rank 1. By setting all the two-by-two minors of M ( P ) equal to zero and simplifying, LIE arrived at the equations y = x 2, z = x 3 which define the normal curve of three dimensional space. Since M ( P ) never has rank 0, H3 leaves no points invariant. Thus the normal curve is the invariant manifold of minimal dimension for H3, and it also follows immediately from these considerations that H3 leaves no linear mani- fo ld - - a point, a line or a plane-- invariant since the invariant manifolds have all been described. LIE also realized that for any n, H,, leaves "nothing planar" invariant [1893: 785], but he apparently never put a p roof in print.

In terms of LIE algebras, the above property of the H,, was recognized as yielding what would nowadays be described as irreducible representations of 31(2). That is, f rom LIE's viewpoint the group PGL(n ~ 1) was " isomorphic" to the special linear group SL(n " 1) [1888: 558], because LIE defined two groups to be isomorphic if their LIE algebras were isomorphic [1888: 292-293]. For PGL(n + 1) and SL(n-+ 1) this is true, as he showed explicitly [1888: 579]. He naturally realized he was using the term " isomorphic" in a sense that did not

52 T. HAWKINS

seem to parallel JORDAN'S definition for finite permutation groups, but he believed he had proved that groups isomorphic in his sense were isomorphic in the more customary sense of a biunique morphism [1888: 419]. One might say that his proof anticipated the theorem that two LIE group germs are locally isomorphic if their LIE algebras are isomorphic. 14 When LIE and his students spoke of groups and their properties, they generally, albeit tacitly, had in mind something akin to a group germ or the corresponding LIE algebra, without of course fully appre- ciating the importance of carefully distinguishing between the local and global properties of groups. Prior to the groundbreaking work of WEYL in the mid-1920's, research on LIE groups was concentrated on the local theory. In particular, when LIE and his students discussed "projective groups in n-dimensional space which leave nothing planar invariant" they were usually discussing subalgebras of 91(n + 1) which leave no vector subspace invariant. 15 Thus whenever it was convenient, LIE and his students would regard the LIE algebra of H n as a subalgebra of fll(n + 1) which is isomorphic to 81(2) and possesses the property that it leaves no vector subspaces invariant. For this reason, we shall frequently use the language and notation of LIE algebras when discussing their work on these projective groups. Using such language, we may say that LIE and his students realized, in effect, that HESSE'S Principle yields irreducible representations of ~(2). In fact, LIE realized that HESSE'S principle yields all irreducible representations of 81(2). That is, he claimed that any three dimensional G ~ PGL(n + 1) which leaves nothing planar invariant must be simple and hence, on the infinitesimal level, isomorphic to dl(2); and that by means of a suitable projective change of variables it can be transformed into H n [1893: 758n].

In the years between 1872, when KLEIN first published his Erlanger Programm, and 1893, when he and LIE finally published their thoughts on the group theoretic aspects of HESSE'S Principle, other mathematicians interpreted the Principle in ways that were ultimately incorporated into the generalizing of the Principle set forth by GINO FANO in his Enzyklopiidie article [1907]. Foremost among them was CORRADO SEGRE (1863-1924), FANO'S teacher. SEGRE obtained his doctorate from the University of Turin in 1883. He began his career at a time that was ripe for the development of the geometry of n-dimensional space. GRASSMANN'S Ausdehnungs- lehre encouraged such a development, especially in the revised form of the edition, of 1862, but most mathematicians ignored it. RIEMANN'S essay of 1854 on space as an n-dimensional manifold, first published as [1868], was perhaps the first in- fluential work in this direction. Then, in the 1870's KLEIN not only published his Erlanger Programm, which expressly championed the cultivation of n-dimensional geometry [1872: Note IV], but also developed PLi3CKER'S ideas on line geometry and in particular stressed the viewpoint of line geometry as the geometry of a quadratic hypersurface in 5-dimensional projective space. A strong additional impetus to the geometrical study of n-dimensional projective space was provided

14 Cf. Corollary 1, p. 280, of BOURBAKI [1975] and his discussion of LIE'S tendency to treat groups locally (pp. 417-421).

15 That these LIE algebras fall within the purview of the KILLING-CARTAN structure theory of semisimple LIE algebras was first pointed out by CARTAN in [1909]. See the discussion of CARTAN'S work below.


by a memoir [1881] on this subject by VERONESE, who published it in the journal KLEIN edited, M a t h e m a t i s c h e Annalen. In particular, many young Italian geometers, including SEGRE, were inspired by it to investigate the geometry of hyperspace which VERONESE presented as a genuinely geometrical science rather than as a dis- guised version of n-variable analysis.~6 SEGRE wrote his doctoral dissertation on line geometry from the Kleinian viewpoint and began to correspond with KLEIN. He was one of the few mathematicians working in the period 1872-1892 who was familiar with the contents of the Erlanger P r o g r a m m and the only one who published a paper expressly aimed at carrying out its ideas. This occurred in a paper [1885] which is of considerable historical importance for the metamorphosis of HESSE'S Principle of Transfer.

The starting point for SEGRE'S paper [1885], was provided by a memoir [1868] by CAYLEV on the problem of studying and enumerating the conics which satisfy a given set of conditions, such as for example the conics which pass through two given points and are tangent to a given conic. Such problems had been studied by CHASLES, who created his theory of characteristics to deal with them. To study such problems CAVLEY introduced a "quasigeometrical representation of conditions by means of loci in hyperspace." That is, i f f (x, y, z) is the general homogeneous polynomial in x, y and z of degree n, its N coefficients a s can be regarded as the homogeneous coodinates of an arbitrary point in (N -- 1)-dimensional projective space. Special curves of degree N are given by f ( x , y , z) - 0 by specifying conditions (equations) that the coefficients a,. must satisfy. These conditions then define a locus in (N -- 1)-dimensional projective space which may be described geometrically. CAVLEY explained that he obtained the idea of quasigeometrical representation from SALMON and referred to SALMON'S papers [ 1866] and [1867]. In these papers SALMON was concerned with the study o fp simultaneous equations in p unknowns for p ~> 3. For p -- 3, the problem can be interpreted geometrically as the intersection of three surfaces in space. "The question now before us may be stated as the corresponding problem in a space o f p dimensions. But we consider it as a purely algebraic question, apart from any geometrical considerations ... . We shall however retain the geometrical language, both because we can thus avoid circumlocutions, and also because we can the more readily see how to apply to a system o fp equations, processes analogous to those which are employed in a system of three" [1866: 327-8].

From CAYLEV'S quasigeometrical point of view, the conic defined in homogeneous coordinates x, y, z by the equation ax z i by 2 + cz 2 ~ 2 f y z + 2gxz r

2hxy -= 0 is to be interpreted as the point in 5-dimensional projective space with homogeneous coordinates (a, b, c, J~ g, h). Thus the totality of all conics in the plane is identified with p5. Within this 5-dimensional space CAYLEV considered the locus of all degenerate conics, i.e., those satisfying the equation abc - - a f z .....

bg z - - ch z _ 2 fgk - 0, obtained by setting the discriminant equal to 0 [1868: 198]. This "discriminant locus" is an algebraic variety of dimension 4 defined by an equation of the third order. Following SEGRE we shall denote it by M2. 111 general degenerate conics consist of two lines in the plane. Among the conics in M2 are those for which the two lines coincide, in this case the equation of the conic

J6 See SEGRE [1891a]. p. 459, and TETRAZINI, [1926], p. 210.

54 T. HAWKINS

takes the form (o~x + fly + yZ) 2 - - 0, and the corresponding points in p5 are given by

(9) (a, b, c ,f , g, h) = (~2, f12, y2, fly, yo~, 0~/3).

These points lie on a 2-dimensional algebraic variety of the fourth order which, again following SEGRE, we shall denote by F 4. This is the surface of VERONESE, who made the most through study of its properties. 17 As CAYLEY realized, these varieties of degenerate conics come into play in the quasigeometrical description of the conics satisfying given conditions.

CAYLEY'S quasigeometrical approach was very congenial to the young SEGR~, who proposed in [1885] to combine it with the ideas of the Erlanger Programm. Since every projective transformation of the plane, i.e. every TE PGL(3), takes

conics into conics it determines a transformation 27 of pS, which takes F 4 into itself and M43 into itself (since T takes lines into lines and line pairs into line pairs)

[1885 : 1 1 ft.]. Moreover the transformations 2? are projective, so that in this manner

a subgroup G Q PGL(6) is defined. Indeed the linearity of T-as a transformation of the homogeneous coordinates a, b . . . . , h was a well known fact from the theory

of invariants, since in effect the transformations 2? are those induced on the coefficients of the quadratic ternary form f - - ax 2 ~- by 2 • CZ 2 + 2fyz + 2gxz + 2hxy by a linear transformation of x, y and z. (SEGRE himself did not

bother to explain why the 2? were projective.) The manifold ps together with the

group G define a geometry in the sense of the Erlanger Programm, and SEGRE proposed to show that "the projective geometry of the conics of the plane coincides exactly" with this geometry, in the sense of Kleinian equivalence [1885: 1].

SEGRE related the ideas in his paper to both the original HESSE Principle of Transfer and to the HESSE-KLEIN Principle. The latter Principle is based on the mapping 7tn of (8) with n = 2, viz. ~2(~x,/3) = (0~ 2, ~x/3,/32) which maps p1 onto the conic (b92 Q p 2 and "transforms" PGL(2) into the group //2 ~ PGL(3) which leaves (g2 invariant. Likewise (9) defines a mapping 7t: p2_> F 4 ~ pS, viz. (9') ~(~,,/3, y) = (0,2,/32, y2,/3y, y~,, ~,/3)

and PGL(3) is transformed into the group G ~ PGL(6) which leaves the surface F~ and the discriminant locus M 3 invariant. As SEGRE put it: "The surface F 4 is transformed into itself by oo s projective transformations of Ss. The projective geometry of the plane coincides with the projective geometry of $5 in which the surface F 4 is fixed, that is if one takes as 'fundamental group '12 the group of the above mentioned cx~ s homographies which transform this surface into itself" [1885: 12]. SEGRE'S footnote 12 refers to the Erlanger Programm where KLEIN used the term "fundamental group".

The connection that SEGRE made with HESSE'S original Principle is more interesting because it is less straightforward. In order to describe the linear varieties in p5 SEGRE found it helpful to set up a one to one correspondence between the

17 See VERONESE [1884]. This paper came to SEGRE'S attention after he had written his own paper [1885].


points of M 3 and pairs of points from the plane. Since each point in M43 represents a degenerate conic, its locus consists of two lines which (by duality) may be regarded as points in p2. (For example, identify the line ~xx + t33, ~ 7 z - 0 with the point with homogeneous coordinates (~, {4, 7)-) Thus we have a correspondence ~ between pairs of points {P, P'} in the plane and points of M43. In particular, the double point pairs (P, P} correspond to points of F4since this surface corresponds to the conics which degenerate into a single line. SEG~E used this correspondence to describe, among other things, the planes of ps that are entirely contained within M43. These turn out to be of two types. The first type, and the one of interest here, corresponds under qb 1 to all pairs [P, P'} such that P and P' lie on a fixed line 5e. Let us denote the corresponding plane in M 3 by lI. SEGRE discovered that F/ intersects the surface F24 in a conic of the plane lI . The points on this conic correspond under qb i to double point pairs {P, P}, P ~ £,a. In this manner, a correspondence between point pairs on a line £a and points in a plane 1[ is established which has the property that the double point pairs correspond to a conic. As SEGRE showed, this is, in effect, the correspondence of HESSE'S Principle of Transfer. "Thus the Uebertragungsprincip of Hesse is again encountered but from a loftier point of view since the correspondene used by Hesse between the points of of a plane and the pairs of points of a line is part of a correspondence between the points of M 3 and the pairs of points of [the plane] . .2 ' [1885: 4]. In a footnote SEGRE added that his derivation of HESSE'S Principle was also more natural: "Moreover, in this manner the principle of transfer no longer appears like an ingenious artifice but as deriving naturally from the fact that the pairs of points of a line [5 a] form a double linear system [/7] of conics ... in which the double points of the line [£0] always form a simply quadratic system."

We noted (following equation (7) above) that the n-dimensional version of HESSE'S Principle tacitly involved the identification of binary forms of degree ,7 with points in n-dimensional projective space. SEGRE'S discussion of HESSE'S Principle in [1885] linked it more explicitly, albeit in a different manner, with CAV- LEY'S quasigeometrical interpretation of forms, in this case ternary forms. Indeed as the above quotations show, SEGRE suggested that a quasigeometrical treatment of HESSE'S Principle was preferable since it provided the Principle with a "natural" derivation from the geometry of 5-dimensional projective space. The quasigeometrical interpretation of forms was to become fundamental to the generalizations of HESSE'S Principle made by SEGRE'S student, FANO, in [1907] and then by CARTAN. In the years following the publication of SEGRE'S memoir [1885] the most outspoken advocate of the quasigeometrical approach was EDUARO SXUDY (1862-1930). STUDY'S work needs to be discussed in some detail because it added new elements to the quasigeometrical approach which were subsequently developed by FANO and CARTAN in their respective generalizations of HESSE'S Principle. in STUDY'S work the quasigeometrical approach to forms became linked with LIE'S theory of groups, the common meeting ground being the theory of invariants. In fact, in this way STUDY became the first to link together the quasigeometrical approach to forms and the representation of LIE groups and algebras, a link whose fertility was demonstrated by CARTAN'S generalization and application of HESSE'S Prin- ciple.

56 T. HAWKINS

Under the influence of GRASSMANN'S Ausdehnungslehre, which he had digested as a university undergraduate, STUDY set himself the task of carrying out research aimed at providing geometrical knowledge with a suitable conceptual and algebraic framework [ENGEL 1931: 140; WEISS 1933: 109]. He was less concerned with per- sonally advancing the frontiers of mathematical knowledge than with the problem of providing an appropriate framework for what was already known. The sixth thesis propounded along with his doctoral dissertation (Munich, 1884) expressed his attitude: "Less effort should be expended on extending the boundaries of the mathematical sciences and far more on regarding what is already at hand from more comprehensive viewpoints." 18 Initially STUDY was of the opinion that GRASS- MANN'S Ausdehnungslehre provided an adequate basis for geometry, but by the time he penned the above words he had become convinced that such a basis was to be found rather in the theory of invariants and particularly in the symbolical method introduced by ARONHOLD and developed by CLEBSCH and GORDAN. The theory of invariants and the symbolical method became, and remained, at the cen- ter of STUDY'S masterplan for mathematical research.

After obtaining his doctorate STUDY moved to Leipzig, where KLEIN was a professor, to write his Habilitationsschrift so as to become an instructor (Privatdo- zent). After deciding not to write this work on the subject suggested to him by KLEIN, STUDY decided to consider CHASLES' theory of characteristics (1864) which dealt with the enumeration of conics satisfying given conditions. CAYLEY'S memoir [1868] discussed above had been inspired by this theory, and during the 1870's several mathematicians gave it their attention since CHASLES had not sup- plied general proofs of his enumeration formula.19 By the time STUDY wrote his Habilitationsschrift [1886] on the theory of characteristics, he had studied the memoirs of CAYLEY and SEGRE and become a staunch advocate of the quasigeometrical interpretation of forms, an interpretation supported as well by the viewpoint of GRASSMANN'S Ausdehmmgslehre. Such an interpretation, explained STUDY, "does not simply serve to provide a much clearer grasp of what is known by taking into account, in the formulation of the theorems, the existing analogies with the relations of ordinary geometry. It also leads to new truths which would be very difficult to discover in another way because even to articulate them without utilizing this analogy would be extraordinarily complicated" [1886: 71].

In 1886 KLEIN left Leipzig for G6ttingen, and SOPHUS LIE left his homeland of Norway to take up the vacant position at Leipzig. LIE came to Leipzig to establish a school devoted to the development, application and dissemination of his theory of transformation groups. Once in Leipzig, LIE saw in STUDY the ideal person to cultivate the connections of his theory with the theory of invariants. Although LIE had called attention to these connections [1884: 545-546; 1893: vii-viii] and discussed them to a certain extent in his lectures on continuous groups [1893 a: Ch. 23], the entire subject was far too algebraic for his tastes. And so LIE turned to STUDY, who already knew something about LIE'S theory from his contact with his close friend FRIEDRICH ENGEL, LIE'S devoted assistant. " I have some

~s Translation of the quotation given by ENGEL [1931: 140]. 19 On the fascinating history of CHASLES' theory up to the present and the place of

STUDY'S work therein, see KLEIMAN [1980].


hope for Study," LIE wrote to KLEIN. "He has become interested in the connection between transformation groups and the theory of invariants. 1 cannot be certain, but 1 have the distinct impression that he will contribute something in this direction ... I am naturally very pleased to have gained capable mathematicians such as Study ...".2°

Sruo¥ ' s contributions to LiE's theory grew out of his quasigeometrical, group theoretic interpretation of the series expansions (Reihenentwickehmgen) that CLEBSCH and GORDAN had introduced into the theory of invariants. Consider, for example, the general binary form of degree m in x and degree n in y:

(1o) f(a; x, ~,)-- ~ ~ ...... ' " .... i),j. aiixl X 2 y 1 i O j 0

Let T denote a nonsingular linear transformation of the x and y variables : x T.v', y = Ty , TC GL(2). Then T transforms f (a ; x, y) into f ' (a ' ; x', y') where

f (a; x, y) = f (a; Tx', T y ) f ' (a ' ; x', y')

and a - + a' defines a nonsingular transformation #(T)C GL(N) of tile N :~ (m + 1) (n + 1) coefficients aij. An invariant of the "base form" f (a ; x, y) is a homogeneous polynomial l(a) such that l(a') -- (det T)"l(a) for all TC GL(2), where w is an integer, or, equivalently, such that l(a') l(a) for all TC SL(2). To anyone inclined to regard mathematics in terms of groups, such as KLEIN or LIE and his students, it was clear that the transformations/~(T) form a group and that ( 2 (T) /~(T l) defines a group homomorphism. The nineteenth century formulation of the notion of an invariant thus leads naturally to certain representations of the special and general linear groups. In particular, if the form

f (a ; x, y) of (10) is of degree 0 in y so that it represents the general binary form of degree m, then the group of transformations e(T), interpreted as transformations of pm, coincides with the projective group H,,, of tile normal curve in pm that arises from HESSE'S Principle, as indicated following equation (8) above.

Suppose now, following STUDY, we interpret the formf(a ; x, y) quasigeometri- cally as a point with homogeneous cordinates aij in N -- l dimensional projective space. Then the [~(T) may be interpreted as projective transformalions of this space (or, of course, as linear transformations of N-dimensional affine space the viewpoint favored by CARTAN). This in itself is not particularly revealing, but it becomes so when applied to the series expansions introduced by CLEBSCH and GORDAN in the 1870's. For the binary fo rmf(a ; x, y) of (10) the series expansion

20 Th.e quotations are from two letters in the archives of the Niedersf.chsische Staats- uncl Universit/itsbibliothek : Cod. ms. F. KLEIN: 10, Nr. 741 (written ca. 1888)arid Nr. 751 (ca. 1889). The passages read as follows: "Zu Study stelle ich. einige Hoffnung. Er hat sich. for den Zusammenhang zwischen Transformationsgruppen und Invariantenth.eorie interessirt, lch kann es nicht sicher beurtheilen : ich h.abe aber den bestimmten Eindruck, dass er etwas in dieser Rich.tung bieten wird. [Nr. 741 ] Es ist for mich natfirlich/iusserst sehr wohl, dass ich ... tfichtige Mathematiker wie ... Study gewonnen habe." [Nr, 751]

58 T. HAWKINS

is an expression of the form l

(11) f ( a ; x, y) ---- ~--u gg(X, y) (x, y)~, k - O

where (x, y) -- x ly2 -- x2Yt and 1 = min {m, n). The forms gk(a; x, y) are polynomials in x and y whose coefficients are expressions in the coefficients aij, whereas the factors (x, y)k do not involve the aij. The g~ were obtained by an explicit "polarization process," easily expressed in the notation of the symbolical method, from forms ek(a; x) in x alone which were called elementary covariants. One of the main reasons the series expansion (11) was deemed important arose from the fact that the invariants o f f ( a ; x, y) coincide with the simultaneous invariants of the elementary covariants ek(a; x). Thus the theory of invariants of forms in two (or more) variable series could be reduced by series expansions to the theory of invariants of forms in one variable series. There were other applications of (1 l) as well [GORDAN 1887: 90-1].

Consider now the quasigeometrical interpretation of the series expansion (11). By letting the coefficients aij in f (a; x, y) take on all possible (complex) values, we obtain N-dimensional affine space, C u. Each term hk(a; x, y) -- gk(a; x, y) (x, y)k of (11) is a polynomial of degrees m and n in x and y respectively with coefficients which depend on the aii. Thus as the aij take on all values the h k describe a subset, Wk, of C u. By virtue of the manner in which the h k are constructed from the elementary covariants e k, they possess properties which, interpreted quasigeometric- ally, translate into properties of W k. For example since the hk turn out to be "linear covariants", Wk is a linear subspace which is taken into itself by the transformations ¢(T). Furthermore, the special nature of these elementary covariants translates into the fact that W k has no nontrivial linear subspaces with this property; the transformations ¢(T)act irreducibly on W~. Viewed quasigeometric- ally, the series expansion (1 l) asserts that CUdecomposes into a sum of irreducible subspaces Wk and yields thereby a complete reducibility theorem for the representation involved. Viewed in modern terms, (11) implies the CLEBSCH-GORDAN series reduction of the tensor product of two irreducible ~[(2) modules of dimensions m 6- 1 and n ÷ 1, respectively. There is, however, a great conceptual gap between the modern interpretation and the actual series expansion (1 l) of CLEBSCH and GORDAN, a gap that STUDY first began to bridge.

STUDY never presented the above interpretation of the series expansion (11) in print because he never had an occasion to expound the theory of binary forms. His publications on the theory of invariants concentrated more on ternary forms, which were of considerable interest for the geometry of the plane. STUDY was the first to publish a systematic treatise on the theory of ternary forms in his book Methoden zur Theorie der terniiren Formen [1889]. Section 12 of Part I[ was entitled "geometrical significance of series expansions" and contained the analogous interpretation for the series expansions of ternary forms. Conscious that such material was rtot customary in a treatise on the theory of i nvariants, STUDY began by justifying its inclusion on the grounds that it made it possible " to perceive conceptually" (begrifflich einzusehen) the theorems on series expansions "or, at least to elucidate them by means of parallel conceptual considerations. "And we do not in the least wish to suppress such a supplement to our treatise since, I believe, we obtain


thereby a deeper sense o f the essence o f the series expansions o f the theory o f in- var iants" [1889: 114].

STUOY'S geometrical interpretation o f series expansions raised two purely group theoretic questions which he discussed with LIE and ENGEL in 1887. 21 Here we express them on the LIE algebra level, the level on which they were in practice considered as noted earlier. Suppose that I C .ql(N) is a LIE algebra with a known structure in the sense that I ~ ,qo, where ,qo is a known Lie algebra. For example, if I is the LIE algebra of the group of t ransformat ions o(T), T { SL(2), associated with the form f(a; x, y) in (10), I m ,qo : ~: ~(2). The first question is: does C u admit a decomposi t ion into linear subspaces Wk with the above described properties ? The series expansions for binary and ternary forms indicated that for any 1 which can arise f rom such forms the answer was affirmative. A theorem positing an affirmative answer for all [ with a given structure will be referred to as a complete reducibility theorem for that structure. The second question arises naturally f rom the first: given a part icular LtE algebra ,qo, what is the nature of the 1 (_ .ql(k) such that i _~- ,qo and I leaves no nontrivial subspace invariant? A theorem providing an answer to this question will be referred to as a characterization theorem for ,qo-

With the assistance of ENGEL, STUDY was able to give a p roof of the complete reducibility theorem for go -- ,~1(2), a l though it was never published [LIE 1893a: 785122. A corresponding characterizat ion theorem was in effect already known to LIE as indicated above: the I with the structure o f ~l(2) which "leave anothing planar invariant" correspond to the projective groups which leave a normal curve invariant. STUDY posited a complete reducibility theorem for .qo = ,~l (3) and worked out a p roo f which, however, was recognized as containing a gap. 23 A correct characterizat ion theorem for ~1(3) was also posited by STUDY [1889" 113] and apparent ly proved to his own satisfaction. 2"~ Following LIE, STUOV'S characterizat ion theorems involved characterizing the I ~ ,qo in terms of a manifold of minimal degree such that 1 is the Lie algebra o f the projective group that leaves it invariant. He also obtained characterizat ion theorems of this sort for .qo 0(4) and conjectured the validity o f a complete reducibility theorem for any semisimple structure, as LIE was very pleased with STUDY'S discoveries and encouraged him to cont inue research in this direction and to publish his results. STUDY, however, never succeeded in obtaining rigorously proved results o f suffi- cient generality to warrant, in his opinion, publication, a l though he did permit

21 The year 1887 for STUDY'S communication of his discoveries and conjectures to ENGEL and LIE is based on STUDY'S recollectio~s in a letter to ENGEL dated 31 July 1892 (F. ENGEL Archive, Mathematisches lnstitut, Justus-Liebig Universitfit Giessen). See also the quotation above from STODY'S letter to ENGEL dated 13 March 1887.

2z The first proof in print is implicit in CARVAN'S doctoral dissertation BOREL [ 1986: 74], although FANO gave what he thought was the first proof [1896: 192ff.].

23 Letter from STUDY to ENGEL dated 31 July 1892 (F. ENGEL Archive, Giessen). S*UDY probably devised his proof in 1887 or thereabouts. See, in this connection, note 21.

2~ In a letter to ENGEL dated 31 July 1892 (F. ENGEL Archive, Giessen). STUDY still regarded this characterization theorem as proved.

25 See STuDY'S letter to LIE dated 31 December 1890 and located in the archives of the University Library in Oslo (Brevsamling Nr. 289).

60 T. HAWKINS

LIE to summarize what he had discovered in his Theorie der Transformationsgruppen [1893: 785-788]. The general problems posed by STUDY were of course extremely difficult, even given the KILLING-CARTAN structure theory of LIE algebras, which was not at STUDY'S disposal. KILLING'S work was published at the time STUDY posed them--and before CARTAN was able to clarify KILLING'S fertile but obscure ideas. It was not until 1924 that WEYL proved the complete reducibility theorem for semisimple LIE algebras, and CARTAN solved the characterization problem for semisimple LIE algebras in 1913. It is perhaps fitting that CARTAN'S solution utilized a generalization of HESSE'S Principle that was based upon the quasigeometrical, group theoretic interpretation of forms that STUDY had advocated.

Before passing on to the further metamorphosis of HESSE'S Principle, it should be noted that STUDY himself took a great interest in the Principle and in fact composed a 160 page essay entitled Das Hessesche Obertragungsprinzip und die In- variantentheorie der Kegelschnittsgruppe. This work was never published. It was probably written at the time he was composing his book on ternary forms for there he pointed out that it was possible to generalize a Principle of Transfer ascribed to CLEnSCH SO as to obtain a principle of transfer with a theory "in which an idea inspired by Hesse is actually implemented" [1889: 178]. 26 The manuscript of the essay appears to have been lost, but judging by its title and the detailed table of contents [given by WEISS 1933:113-114], STUDY embedded HESSE'S Principle in the group-theoretic type of invariant theory he presented in his book on ternary forms.

The first application of STUDV'S complete reducibility theorem and LIE'S characterization theorem for ~(2), was made by SEGRE'S student GINO FANO (1872-1952) in [1896]. FANO commenced his mathematical studies at the University of Turin in 1888, the same year that SE~RE was appointed to a professorship there, and from the beginning they were in close contact. It was SEGRE who acquainted FANO with KLEIN'S Erlanger Programm. In the 1880's the Erlanger Programm was still not well known or accessible, having appeared in 1872 as a "Programm- schrift". Feeling that young Italian geometers were not familiar with its contents, SE~RE encouraged FANO to publish an Italian translation (1890). FANO also spent the academic year 1893--1894 in G6ttingen with KLEIN, and during the 1890's he published a number of papers on geometrical applications of LIE'S theory of transformation groups. For example, in the memoir [1896] referred to above, FANO dealt with the problem of determining all varieties in p3 left invariant by some (infinitesimal) projective group. 27 He was thus a natural candidate to write the article [1907] on continuous groups and geometry for KLEIN'S Encyklop6die tier mathematischen Wissenschaften.

FANO gave a prominent place to the HESSE-KLEIN Principle of Transfer in his encyclopedia article. Part II of the article, which dealt with KLEIN'S notion of equivalent geometries, featured the HESSE-KLEIN Principle to exemplify this notion,

26 CLEBSCH'S Principle was presented in [1861] but was first characterized as a principle of transfer in LINDEMANN'S edition of CLEBSCH'S lectures [1876: 274]. These lectures also claim to contain, as a special case of CLEBSCH'S results or~ algebraic curves, the "simplest" algebraic expression of HESSE'S Principle [1876: 887n].

27 For further details, see HAWKINS [1984: 452ff].


first for n = 2 with a brief reference to HESSE'S original approach and then for arbitrary n [1907: 358-362]. Then, with the remark that the underlying idea of the Principle "amounts to representing the linear system of all binary forms of order n by the points (or by the [hyperplanes] Rn -0 of R , , " FANO observed that this idea "can be applied to every linear system of algebraic forms" [1907: 362]. That is, let f ( x ) , i -- O, 1, . . . , n, denote homogeneous polynomials of degree d in Xo, X~ . . . . . x~, k ~ n . Then

(I 2) f (2 ; x) = ~2 2if/(x) i 0

is a linear system if for all T E GL(k 4- 1) and x -- Tx', it follows that

(13) f(2; x) = f ( 2 , Tx') = f0 . ' ; x ') ,

which means that the f i (x) have the property that

(14) T" f (x) =-~ f (Tx) = ~ aijfj (x). j 0

In other words, the fact that f (2 ; x) forms a linear system meant that it was possible to develop a theory of invariants with respect to the system, an invariant being a homogeneous polynomial/(2) such that /(2') ~ (det T) ~ I(2) for all TE GL(k + 1). (Of course (14) interpreted in present-day terms means that the linear system forms a module for GL(k 4- 1).) As FANO observed, there are then two ways to identify this system of forms with points in pn. One way is to identify f (2; x) with (20, . . . , 2,,); the other is to regard f(2; x ) = 0 as a bundle of hyperplanes--one for each choice of the 2i--defining the common point (fo(X) . . . . . f , (x) ) . FANO emphasized the latter interpretation since it yields the mapping ~P : ( Xo . . . . . xk) -*- (fo( X) . . . . . fn( x) ) from pk into P". Setting Mk = ~v[p*], FANO could then declare in the spirit of the Erlanger Programm that " the theory of invariants of the linear system [f(2; x)] coincides with the projective geometry of this Mk." As FANO observed, if k -- 1 and f (2 ; x) is the general binary form of degree n, then the mapping 7 t is that of the HESSE-KLEIN Principle of Transfer described in (8) and Mk is the rational normal curve (g,. Likewise, as he also observed, if k -- 2 and f(2; x) is the general ternary form of degree two, then 7 t is the mapping defined by (9') so that Mk is the VERONESE surface Fz4. 28

FANO had thus given a broad generalization of the HESSE-KLEIN Principle which included the mappings of KLEIN, SEGRE and VERONESE as very special cases. He did not stop there, however, but went on to observe that the linear system could just as well involve homogeneous polynomials in several variable series (Xo . . . . , Xp), (No . . . . . Yq) . . . . . For example, in the case of two series f(2; x, y) -----

~2 2if/(x, y), and one has the mapping ~u: Ix, y} -+ (f~(x, y) . . . . . . f,,(x, y)) from i 0

P P x P q into pn. FANO made this further generalization of the HESSE-KLEIN Prin- ciple so that he could embrace within its scope SEGRE'S memoir [1891] on what

28 For any k, if f(2; x) is the general form of degree d, ~P is now usually called a VERONESE mapping and M k a VERONESE variety.

62 T. HAWKINS

are now called SECRE maps and varieties. Since geometers study various corres- pondences between points x E PP and y ~ p,t, such as projectivities and corre- lations, SEGRE had proposed identifying point pairs so related with geometrical objects in a higher dimensional projective space. He observed that if x = (x o . . . . . Xp) and y = (No . . . . . yq ) then the pair {x, y} may be identified with the point in pX, N = (p + l ) (q + 1 ) - 1, with homogeneous coordinates Xja ~: xjy/. In this case, the forms f ( x , y) defining the linear system are the xj)'k, and Mp~,f gt[pp ~ pa] is the var ie ty- -now called a SECRE variety--within which the proposed geometrical study of the correspondence between x and y takes place. As with the quasigeometrical study of forms, SE6RE regarded this approach within the framework of the geometrical theory of hyperspaces inaugurated by VERONESE [1891: 174, n. 1]. SEGRE made a detailed study of the case p = q 2 as a paradigm for the case p -- q and stressed the importance of these cases not only for the study of the plane and ordinary space but also for the study of the representation of points of a complex projective space in a real projective space. 29

Although FANO stressed the mapping aspect of his generalizations of the HESSE- KLEIN Principle, they also had a group theoretic import which CARTAN subsequently emphasized and developed. For example, corresponding to the linear system

]'(2; x) of(12) is the representation T - + or(T) of GL(k ? 1) defined by (14) and also the contragredient representation yielded by the transformation 2 ' : /*(T)2 of (13), namely T--+ ~(T) / ,(T -~) - a(T ~)*, where t denotes matrix transposition. These representations are implicit in FANO'S Kleinian declaration quoted above. The same is true in the case of a linear system in several variable series such as f0 . ; x, y). When p = q = k the " theory of invariants of the linear system" ]'(2; x ,y ) implies the representation T--> ~r(T) of GL(k _L 1) defined by

(15) T..f~',x, y) /i',Tx, Ty)

as well as the corresponding contragredient representation. When p ~ q, representations of GL(p q- 1)>< GL(q + 1) are involved. Thus FANO'S generalizations of the HESSE-KLEIN Principle of Transfer could be interpreted as principle for generating representations, and this is the interpretation that CARTAN chose to extend even further.

ELIE CARTAN (1865 1951) was one of several graduates of the prestigious Ecole Normale Sup6rieure who were encouraged by the leaders of the French community of mathematicians (PICARD, DARBOUX, POINCAR~) to develop LIE'S theory of transformation groups and, above all, its applications to the study and classification of differential equations. In his doctoral dissertation [1894], CARTAN took on the formidible task of making rigorous mathematical sense out of the ground breaking papers on the structure of semisimple L1E algebras written by

29 The real representation of complex points was the concern of SEGRE'S theory of hyperalgebraic varieties [1890, 1892] which, like FANO'S generalization of the HESSE- KLEIN Principle, was to provide CARTAN with useful ideas for his study of the representation of LIE algebras - a further indication of the extent to which his trilogy of memoirs [1913, 1914a, 1914b] were a byproduct of his geometrical interests at the time he was revising FANO'S encyclopedia article [1907].


WILHELM KILLING in 1888-1890. 30 In particular he confirmed the validity of KILLING'S classification of simple LIE algebras which was considered especially important by virtue of its relevance to the GALOIS type theory of differential equations envisioned by LIE. Most of CARTAN'S subsequent research involved the further development of LIE'S theory along lines that gave prominence to structural considerations. CARTAN probably first became acquainted with HESSE'S Principle through his work on the task of preparing a revision of FANO'S article [1907] for the French edition of the Eno, clopiidie. His acceptance of this task was indicative of a general shift in his research interests towards more geometrical aspects of LIE'S theory. Precisely when he began working on the encyclopedia project is unclear, but certainly by the time he composed his trilogy of memoirs [1913, 1914a, 1914b]. geometry and geometrical applications of groups along the lines discussed by FANO were foremost in his mind. Indeed, he did much more than simply translate FANO'S 97-page article into French. Over 37 pages of new material were added to the French edition. Thirteen of these were devoted to a further discussion of HESSE'S Principle. After presenting, more or less verbatum, FANO'S presentation of the HESSE-KLEIN Principle of Transfer and FANO'S extensions of it, CARTAN launched into his own further extensions under the heading "The principle of Hesse applied to the generation of projective groups homomorphic to a given projective group" [1915: 99].

CARTAN'S generalizations were based upon the simple observation that in FANO'S extensions of the Principle, there was no need to restrict onself to the group of all projective transformations. That is, let G denote any group of projective transformations o f P ~. CARTAN regarded G as a subgroup of GL(k + 1) which is legitimate in the sense that, at this time, he still thought about transformation groups on the same local (or "group germ") level that permeated LIE'S work, as discussed above. This must be kept in mind in the ensuing discussion of CARTAN'S work and the term "group" understood as something akin to a group germ. Given G ~ GL(k + 1), the forms f(2; x) of (12) may be said to constitute a linear system if the analog of (14) holds:

(I 4') r" f~(x)~ ~ f (Tx) - ~, c;iifj(x) VT ~ G. i 0

In this way a representation of G of degree n + 1 is obtained, or, as CARTAN put it: the projective group G of pk thus gives birth to a homomorphic projective group of P". CARTAN generalized FANO'S ideas relating to (15)in two directions. If G~ ~ GL(p + 1) and Gz ~ GL(q 4- 1) are both homomorphic images of the same transformation group H, with T1 and Tz corresponding, respectively, to

T~ H, then f (2 ;x , y ) -~ 2 2 ; f (x ,y ) may be defined to be a linear system if i 0

the following modified version of (15) holds:

(15') T.f~(x,y) f (T , x , T2y) ¥T~ H.

30 The ensuing discussion of the work of KILLING and CARTAN on the structure of semisimple LIE algebras is based upon HAWKINS [19821. The discussion of CARTAN'S work in the period 1894-1915 is based upon the more extensive analysis and documen- tation in HAWKINS [1984a].

64 T. HAWKINS

In this manner a repesentation of H is obtained. If GI and Gz are homomorphic images of nonisomorphic groups H1, //2, and if

f (a ; x) = ~ 2if(x), g(tt; y) = ~ tt.igj(y) i--O j--O

are linear systems for G~ and G2, respectiveIy, then

h(v ; x, y) = "~ ~ vi.ifi(x) gj(y) i=0 j = 0

defines a linear system for GI × G2 and

(16) (T× T') . f ( x ) gj(y) -~= f ( r , x ) gfiT2y)

definesahomomorphism of HI×H2, where T--~ T1 and T'---> T2 bythehomo- morphisms HI ~ G~, H2 ~ G2. By virtue of (14'), (15') and (16), HESSE'S Principle was thus transformed by CARTAN into a principle for manufacturing representations. Citing his memoir [1913], he declared that from the extended Principle it was possible to solve the following problem: " to determine all projective groups which leave no planar variety invariant" [1915: 100].

CARTAN had already encountered this problem in a different mathematical guise when he was working on his doctoral dissertation [1894]. There he had considered the following problem: Let g be a nonsolvable LIE algebra with radical r and let ~ denote the semisimple LIE algebra g/r. Determine up to isomorphism all such g which satisfy two additional conditions: (a) g is indecomposable in the sense that it is impossible to express g as a vector space sum g = g~ + gz of two subalgebras such that [g~, g2] = 0; (b) ¢ is the only nontrivial solvable ideal in g (and therefore [r, t] =- 0). This problem, with some unnecessary re- strictions, had been formulated by KILUNG in his work on the classification of Lm algebras (1889:116). He claimed, without any justification, that its resolu- tion would constitute the major step towards the classification of all nonsolvable LIE algebras. We shall refer to this problem as Killing's Problem. CARTAN'S interest in KILLING'S Problem was due to the fact that he realized its equivalence to the following problem: determine all finite-dimensional semisimple groups of linear, homogeneous transformations in n variables which leave no "planar manifold" invariant [1894: 134]. We shall refer to this problem as Cartan's Problem. The connection between the two problems derived from the fact that if g is a Lm algebra described in KILLING'S Problem, and if l ( g l ( r ) consists of all linear transformations T x, XC ~, defined by T x ( Y ) = [X, Y] for all Y in r, then condition (a) above implies that X-+ Tx is a LIE algebra isomorphism, and (b) implies that 1 leaves no vector subspaces invariant. CARTAN'S Problem was of interest to CARTAN because he perceived the value for applications of a linear representation of a LIE algebra. For example, having confirmed KILLING'S classification of simple Lm algebras, he deemed it of "extreme importance" for the envisioned GALOIS theory of differential equations to obtain, for each simple structure type, the "simplest" concrete representation [1894: 135]. This meant solving the following problem: For each simple structure type, construct the solution to CARTAN'S Problem for which n is minimal. That is, determine for each simple


LIE algebra the nontriviat, irreducible representation of minimal degree. We shall refer to this as the Minimal Degree Problem.

For C•Ra'AN, the significance of the equivalence of the problems of KILLING and CARTAN was that, in dealing with the latter problem, he could utilize the tools introduced by KILLING to deal with the former, namely the structure theory of semisimple LIE algebras and the theory of "secondary roots" (Nebenwurzeln). By correcting KILLING'S misconceptions about secondary roots (especially regarding their multiplicities)and by further developing the ideas involved, CARTAN ended up creating the modern theory of weights. The process was begun in his dissertation and completed in his memoir [ 1913]. The connection between weights and KILLING'S secondary roots can be explained readily using current terminology and the notation introduced in KILLING'S Problem. As indicated above, r may be regarded as an d-module. The weights of this module correspond to the secondary roots of ~. For the purposes of this essay it will suffice to use current terminology in discussing what KILLING knew about secondary roots. He realized, for example, that if S denotes the set of weights (secondary roots) associated with a LIE algebra .q described in KILLING'S Problem, then there exists a weight

~ S such that S is the smallest saturated set of weights containing ~. As in the modern theory, KILLING considered weights of ~ formally, i.e. as certain linear combinations of the roots of 8. Then to a weight ~ there corresponds the smallest saturated set S(~) containing it, and KILLING'S Problem suggests asking whether a corresponding ~q actually exists, i.e. whether an g-module ~ exists. KILLING believed the answer was in general negative. For example for d simple of type B his misconceptions about the multiplicities of weights apparently led him to conclude that many S(~r) exist for which there can be no corresponding r [1889: 120].

In [1913] CARTAN solved his general Problem by proving that for every saturated set S(.~) a corresponding r does in fact exist. In his dissertation he had solved the Minimal Degree Problem by showing that for each simple type, there were one or at most two S(z 0 whose cardinality made them candidates for corresponding to the representation of minimal degree. Multiplicity considerations narrowed the candidates to one per simple type, and he then proceeded to construct the corresponding representation. Although he suppressed the calculations involved, they must have been considerable. 31 CARTAN had achieved a similar feat in con- firming KILLING'S classification of simple LIE algebras. For the exceptional types KILLING specified what amounts to the CARTAN matrices, but, with the partial exception of type G2, he did not construct examples of these types although he was convinced they existed. CM~a'AN proved they existed by a type-by-type construction of examples. The solution of CARTAN'S Problem in general, however, could not be achieved in this fashion since it was clear that, even for a fixed simple type with a specific rank l, the number of S(~r) was infinite. It was in this connec-

31 In his memoir on the classification infinite-dimensional simple groups, CARTAN also had to construct representations of finite-dimensional simple LIE algebras which corresponded to certain S(:0. On that occasion he wrote: "Nous n'entrerons pas dans le d6tail des calculs qui conduissent ~. cette d6termination, ces calculs 6tant assez longs et fastidieux" [1909: 150].

66 T. HAWKINS

tion that the ideas underlying HESSE'S Principle as extended by CARTAN proved especially useful.

CARTAN'S interest in HESSE'S Principle and his commitment to revise FANO'S encyclopedia article [1907] were aspects of his increasing interest in geometrical applications of LIE'S theory, an interest which became prominent after he solved the classification problem for infinite-dimensional simple groups [1909]. These new interests also heightened his interest in solving CARTAN'S Problem. He sus- pected that every geometry in the sense of KLEIN'S Erlanger Programm was equivalent to a projective one, that is, one defined by a projective group [1915: 92]. This suspicion was based upon his conjecture, eventually confirmed by ADO'S Theorem [1934], that, given any finite dimensional LIE algebra, a projective group exists with that LIE algebra as its infinitesimal group [1908: 449n, 249]. The problem of determining all projective groups was therefore important because its solution, assuming the validity of the conjectured theorem, would yield all possible geometries corresponding to finite-dimensional groups [1914c: 541-542]. By virtue of a theorem first stated in his work on infinite-dimensional simple groups [1909: 148], CARTAN realized that the problem of determining all projective groups which leave no "planar manifold" invariant--in essence SXUDV'S Characteriza- tion Problem--was equivalent to his Problem. That is, he realized that any LIE algebra 1 Q ,ql(n) with the property that it leaves no vector subspaces invariant is either semisimple or of the form 1 ~ It + {21}, where 1~ is semisimple. In [1913] he solved STUDV'S Characterization Problem for complex projective groups by using the theory of weights to solve his Problem, and in [1914b] he solved the analogous problem for real projective groups, after first classifying real semisimple LIE algebras in [1914a].

To illustrate the way CARTAN utilized the ideas behind his version of HESSE'S Principle, we first consider his determination of all simple I Q ill(n) which leave no vector subspaces invariant. Corresponding to each simple structure type of rank l, he showed how to define dominant weights zt t . . . . . nt, which he termed fundamental because any dominant weight n could be expressed as a nonnega-

l

tive integral linear combination ~ = ~ kiwi [1913: 66]. For each :7~ i CARTAN i - - l

produced a corresponding representation 1i, just as he had earlier produced a LIE algebra and a minimal degree representation for each simple type, that is, by means of a case-by-case construction. HESSE'S Principle then yielded a represen-

t tation corresponding to z~ = ~ kin,-. For example, to obtain a representation

i l

corresponding to n = zq + n2, assume the variables Xo, ... , Xp and Yo . . . . . Yq on which the transformations of 1~ and 12 act are chosen so that x 0 and Yo are highest weight "vectors" for nl and Jr2, respectively, and let LI C GL(p + 1), L2 ~ GL(q -k l) be the corresponding groups (or group germs). (The transition from LIE algebras to groups and back again is made tacitly in CARTAN'S memoir.) CARTAN now introduced a new way of defining a linear system which is somewhat reminiscent of the way FANO interpreted SEGRE'S mapping in terms of HESSE'S Principle. Let fo(X, y) -- XoYo and consider all the forms generated from fo by applying all corresponding transformations T,, 7"2 of L~, L 2 to fo "successively


and repeatedly" in accordance with (15'). Then f l ( x , y ) . . . . . £ ( x , y ) may be chosen f rom among them so that fo . . . . . fn are linearly independent. Hence

f (2 ; x, 3') -- £ 2~ f (x , y) is a linear system. The representation defined by this i 0

system corresponds, on the infinitesimal level, to the CARTAN product representation for ~L + a"g2 "32

CARTAN also showed that HESSE'S Principle could be used to great advantage in dealing with the problem of determining the "fundamental groups" L~ associated with the fundamental dominant weights ~r~ [ 1915 : 101 f. ; 1913 : 32 f.]. This provided further occasion for him to link his work with the geometry of the nineteenth century. A basic tenet underlying the Erlanger P r o g r a m m and the geometrical work of LIE was that the "elements" which are coordinatized in describing a geometry need not be literal points. This view was inspired by PL0CKER'S introduction of line coordinates in space in 1846. PL0CKER'S ideas were modified to suit the needs of projective geometry by CAYLEY in 1862, and in that form they were applied extensively in geometrical research by both PL/3CKER and KLEIN, his student. 33 In [1872] CLEBSCH extended the idea of line coordinates to P" and further generalized it as follows. If Xo, Xl . . . . . x,, are homogeneous coordinates

(n ~ 1) x i X][associated in P", then one may consider the , 2 line coordinates x i j Yi J'j

with the line determined by x and y. Likewise. one may consider t he ( n ~ 1] \ 3 / plane coordinates

Xi, j , k

X i X j X k

5'~ 3'j 3k

z i zj z~

associated with the plane through x, y and z, and so on up to the ( n i 1] coordin- \ II /

ates of a hyperplane. With geometrical applications in mind, CLEBSCH stressed the need to study the invariant theory of forms in all these types of variables. In his book on ternary forms [1889] STUDY did just this in the case n =- 2.

The most commonly considered cases were, in fact, n 2 and n = 3 since they corresponded to the geometry of the plane and space. Consider, for example the case n - - 3. Since a projective point t ransformation of p3 also transforms lines, each such projective transformation of p3 induces a transformation of its lines and, more generally, a t ransformation of the line coordinates xi, j (since the x i j of an actual line must satisfy a quadratic equation). In this manner the group of projective transformations of p3 induces a group of transformations of its lines. This way of thinking was commonplace for LIE and his students. CARTAN pointed out that it could be interpreted as a special case of his version of HESSE'S

32 CARTAN took it for granted that this representation is irreducible, although th.is is not immediate. On this point, see the remarks by BOREL [1986: 57, 73n. 8].

33 For further details and references regarding the development and application of PLOCKER coordinates and their relation to GRASSMANN'S ideas, see FANO [ 1907 : 310, n. 62 ].

68 T. HAWKINS

Principle and that, so conceived, it provided a way to create systematically new groups from known ones. Consider, for example, SL(l + 1), which, following CARTAN, we regard as defining the projective transformations of/-dimensional

projective space. In P' t h e ( l + 1) k coordinates xij,.., are forms of degree k in

X 0 . . . . . X l and f(2; x ) = ~ , i , L . . . xi, j .... defines a linear system and hence a representation L k of SL(l 4- 1) for k = 1, 2 . . . . l. (For k = 1 the coordinates are the x i themselves and L~ = SL(I 4- 1).) CARTAN showed that L k is the "fundamental group" corresponding to the fundamental dominant weight ~k and highest weight vector x0,1,...,k-~. Thus to construct the representation L corresponding

l

to ~ = Y~ pk~rk, let k = l

(17) fo = ~ .p l~ .p l Pl ~ 0 "~0,1 ' ' - X 0 , 1 , . . . I - I

and apply all the transformations TE SL(I 4- 1) to generate the linear system

f(2; x) = ~ 21f/(x) yielding L by HESSE'S Principle. i - -O

Nowadays these constructions are carried out within the conceptual framework of tensor algebra as indicated in the introductory remarks. In particular, the space

of (k --1)-dimensionaJ planar manifolds in Pt coordinatized by the ( l 1) general-

ized PLi3CKER coordinates xi3,.., would be replaced by the skew symmetric tensors in a k-fold tensor product of (l + 1)-dimensional vector spaces. But this was not CARTAN'S viewpoint in 1913-15. He took great delight in finding points of contact between the abstract considerations into which structural considerations led him and the geometry of the nineteenth century. HESSE'S Principle afforded him one such contact point. Indeed, as CARTAN observed, if I---- 2 and the fo of (17) is taken as f0 = x2 (and thus ~ = 2z~0, then the linear system generated by applying the transformations of SL(3) tofo is the linear system of ternary quadratic

2

forms f(2; x) = ~ 2ijxixj. Thus "one obtains in the space E 5 [viz. ps] the pro- i , j = O

jective group which is isomorphic to the projective group of the plane and leaves the variety 3//3... invariant" [1915 : 102]. The variety M43 is the variety of degenerate conics studied by CAVLEV in [ 1868] and by SEGRE in [1885], and the group constructed

by CARTAN is the group G underlying SEGRE'S approach. (See the remarks above following (9').)As we saw, SECRE used the quasigeometrical setting yielded by the above linear system as a context for deriving HESSE'S Principle, as well as the HESSE- KLEIN Principle. CARTAN in a sense turned the tables and used HESSE'S Principle

to derive from it the Kleinian framework of SEGRE'S memoir: the group (~ defining the geometry. Of course what CARTAY meant by HESSE'S Principle was quite different from what SEGRE had in mind. The quasigeometrical framework that SErRE had used to derive HESSE'S Principle "naturally" had, through the intermediary of STUDY and FANO, become a fundamental aspect of the new interpretation of the Principle presented and applied by CARTAN.


In 1866 when HESSE presented his Principle of Transfer to his fellow mathematicians, it was conceived as an analog of the Principle of Duality. The idea was to set up a correspondence between different types of geometrical objects in such a manner that theorems about the one class of objects would transfer automatically to theorems about the other class. KLEIN realized that this idea was closely related to his own notion of differing modes of geometrical treatment and their transference under mappings. It is even conceivable that HESSE'S work helped inspire KLEIN to articulate the ideas of his Erlanger Programm. In any case, by bringing HESSE'S Principle within the conceptual framework of his Erlanger Programm, it became linked with the theory of transformation groups that LIE had begun to work out and, more specifically, with what was to become the representation theory of LIE algebras: The groups Hn associated with the HESSE-KLEIN Prin- ciple of Transfer yielded, as LIE observed, all the irreducible representations of ~1(2). The n-dimensional setting of HESSE'S Principle [MEYER 1883] was also congenial to CAYLEV'S quasigeometrical interpretation of certain systems of forms (later characterized as linear systems by FANO), and this interpretation was emphasized by SEGRE [1885] and by STUDY [1886, 1889]. SEGRE claimed that the quasigeometrical interpretation of quadratic ternary forms provided the natural setting for HESSE'S original Principle while the HESSE-K•EIN Principle of Transfer (8)

generalized to a mapping (9') yielding the group G and thereby an irreducible representation of dl(3), as CARTAN observed. STUDY advocated the quasigeometrical interpretation of forms on a grander scale. He saw it as essential for a conceptually oriented treatment of the theory of invariants, and it was from this conceptual viewpoint that he sought to realize LIE'S hopes of a fruitful liason between the theory of invariants and the theory of groups. In this manner STUDY was led to grapple with problems of fundamental importance to the nascent theory of the representation of DE algebras--his complete reducibility and characterization problems--and it was in this connection that the quasigeometrical interpretation of forms was linked again with HESSE'S Principle : the solution to STUDV'S characterization problem for dl(2) was provided by the groups Hn associated with the HESSE-KLEIN Principle of Transfer. FANO [1907] accepted STUDV'S quasigeometrical approach to the theory of invariants and declared the underlying idea of HESSE'S Principle to be the idea that the linear system of binary forms of degree n can be identified with pn. So interpreted, FANO proceeded to extend HESSE'S Principle to other linear systems, thereby systematically linking invariant theory with geometry and bringing within its scope the mappings of VERONESE and SEGRL. FANO'S generalization of HESSE'S Principle thus set the stage for the further generalizations that were made by CARTAN and that were motivated by his interests in the group-theoretic implications of the Principle when viewed in the light of the problems and methods of LIE'S theory: KILLING'S Problem and the theory of secondary roots (or weights). By 1915 the connection of HESSE'S Principle with the Principle of Duality was far from apparent. What had originated as a device for generating geometrical theorems had became a device for generating irreducible representations of semisimple LIE algebras.

Postscript, The objective of the above essay was to unravel the historical thread connecting HESSE'S original Principle with CARTAN'S version, to trace the meta-

70 T. HAWKINS

morphosis of the Principle from HESSE to CARTAN. One question consequently ignored in our essay is: Did the study of FANO'S Encyklopiidie article and particularly his discussion of HESSE'S Principle inspire CARTAN'S idea of the CARTAN product construction, or did he subsequently see that this construction could be interpreted in terms of HESSE'S Principle as set forth in the French edition [1915] of FANO'S article? To my knowledge there is no documentary evidence that provides an unequivocal answer to this question. What evidence there is indicates that when CARTAN wrote his trilogy of memoirs [1913, 1914a, 1914b] he was quite familiar with the corpus of geometrical research reported in FANO'S article. For example, to solve the problem of determining all the real, irreducible representations of simple LIE algebras over R in [1914b], he utilized ideas from SEGRE'S theory of hyperalgebraic varieties [1890, 1892] a4. And he made a point of showing that special cases of his theory yielded " f rom another point of view" well known geometrical results such as RIEMANN'S representation of complex numbers on the sphere [1914b: 182-183] and the representation of the rotations of space by the

homographies z-+ (az + b)/(--bz + ~) [1914b: 180], which KLEIN made fa- mous in his lectures on the icosahedron [1884]. And, although CARTAN did not mention HESSE'S Principle by name in [1913], he pointed out that a special case of the CARTAN product construction was already well known: "The results obtained thus include those already known relative to the generation of three-parameter projective groups which leave no plane multiplicity invariant: each of them indicates in effect how the projective group of the line transforms the expressions

X~, xP--Ix2 . . . . . XIX~ -1, XP9

among themselves" [1913: 2]. This amounts to a description of the HESSE-KLEIN Principle of Transfer (8) and the associated group lip. Thus it is clear that when he composed his trilogy of memoirs, the connections with the geometrical researches reported on by FANO were very much on CARTAN'S mind. Indeed the above quotation suggests that CARTAN perceived the connection between HESSE'S Principle as discussed by FANO and his product construction. But the question as to whether the Principle inspired the product construction seems destined, like many such questions, to remain forever unanswered.

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Department of Mathematics Boston University

(Received May 18, 1987)

hesses's principle of transfer and the representation of lie algebras

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