Book Review

Emergence of the of Lie Groups: An Essay in the History of , 1869–1926

Reviewed by David E. Rowe

Emergence of the Theory of Lie Groups: earliest work appeared, An Essay in the , 1869–1926 up to Weyl’s classic pa- Thomas Hawkins pers from 1925 to 1926, New York, Springer-Verlag, 2000 along with a clearly 564 pages, $79.95 stated goal. In the pref- ISBN 0-387-98963-3 ace he calls his book “both more and less Among historians of mathematics Thomas than a [comprehensive] Hawkins has long been regarded as the leading history of the theory of authority on the early history of Lie groups. After Lie groups”. Less, be- decades of work on this fascinating theme, he has cause it deals primarily synthesized the fruits of his labors in this 564-page with the origins of the Essay in the History of Mathematics, a landmark theory and those parts study that represents the culmination of his contri- of it pertaining to the butions to the field. In recognition of these accom- structure of Lie groups plishments, the AMS recently named Hawkins its and Lie . More, first recipient of the prestigious Whiteman Prize because he has a broader agenda in mind that goes beyond the challenge of weaving together the for exceptional scholarly writing on the history threads of this complicated story. of mathematics (see the citation and Hawkins’s Historians, at least traditionally, are supposed response in Notices 48 (2001), 416–7). Having to be good storytellers, but serious history usually followed his work for many years, I would like to has a larger purpose in mind. Crudely put, histo- take this opportunity to congratulate the recipient rians try to understand the past by explaining why and to commend the AMS committee that nominated certain things happened the way they did. Hawkins him for this award; Hawkins richly deserves this high set himself the challenge of trying to understand distinction. The committee’s citation also somewhat what motivated the principal actors who con- relieves me from the strain of finding appropriate tributed to what eventually became known as the superlatives to describe the present book, which theory of Lie groups. He wants to tell us not just requires no further personal endorsement. what happened but why, by showing how the rel- Hawkins’s “essay” sets sharp temporal bound- evant mathematical developments were influenced aries spanning the period from 1869, when Lie’s by the specific contexts in which this work was un- dertaken. What makes his book such impressive David E. Rowe is professor of the history of mathematics reading is that he never loses sight of this goal, and natural sciences at Mainz University. His email address despite all the obstacles involved, including the is [email protected].

668 NOTICES OF THE AMS VOLUME 50, NUMBER 6 technically demanding mathematical ideas he experience tells me that are not recounts so faithfully. generally accustomed to thinking about antiquated Hawkins dives headlong into many facets of re- results and methods. This strangeness poses a search activity in surrounding areas that had a direct major challenge for Hawkins’s readership beyond bearing on the rather sporadic work that eventually the technical competence his book demands. Histo- yielded the structure theory of Lie groups and alge- rians after all are expected to engage their subject bras. Those who take the plunge with him will learn matter on its own terms, whereas mathematicians much about the mathematical worlds in which some strive to move forward on the basis of sharply of his main actors moved. He teaches us about the circumscribed, state-of-the-art knowledge. Those influence of Jacobi’s ideas on Lie’s nascent theory familiar with both mind-sets know that they are at of groups and the impact of Weierstrassian ideas on least potentially antithetical, and as a historian of ; we learn about the Parisian response mathematics I feel compelled to emphasize that we to Lie’s theory, especially as revealed in Élie Cartan’s should “mind the gap” that separates them. Thus, work, and the broader impact of the Göttingen at- while I agree with Hawkins that a study like the pre- mosphere and how this shaped ’s sent one can surely lead to a deeper understanding approach to global theory of Lie groups. The book’s of contemporary mathematics, I doubt that even ex- four-part structure based on the contributions of perts on modern Lie theory will find this a sufficient Lie, Killing, Cartan, and Weyl helps lend conceptual motivation to read his book from cover to cover. clarity to an undertaking of truly vast scope. Some- So prospective readers be forewarned: this is a how, despite his penchant for technical and histor- full-blooded historical study about the emergence ical detail, the author manages to keep the main of the theory of Lie groups. It is chock full of twists storyline clearly visible, even as he finds himself and turns, with plenty of mathematical dead-ends spinning ever finer webs of mathematical ideas and and, above all, unfamiliar ideas. The fellow who first intellectual contexts, culminating with a brilliant dis- got the ball rolling, (1844–1899), was a cussion of the motivations behind Hermann Weyl’s visionary figure and saw himself in just that light. work on the representation theory of Lie groups. Writing to the poet Bjørnstjerne Bjørnson in 1893, This is a book that opens new vistas for histo- Lie likened himself to the Norwegian poets of the rians of mathematics, and for this reviewer it offers day: “…without fantasy one would never become the opportunity to reflect not only on what Hawkins a , and what gave me a place among has accomplished but also on what remains to be the mathematicians of our day, despite my lack of done in the light of his achievement. In what knowledge and form, was the audacity of my think- follows I will try to survey the terrain he covers, ing.”1 One can scarcely exaggerate the enormous with an eye to some important themes and ques- differences between Lie’s original ideas and the tions that deserve further consideration. Before modern theory that bears his name. In fact, the doing so, however, a few prefatory words should terminology of Lie groups and Lie algebras became be said about this volume’s prospective audience. current only in the 1930s, more than thirty years In the preface Hawkins writes that his book is in- after Lie’s death. Lie’s original theory dealt with tended for students of mathematics as well as transformation groups, which were closely akin to mathematicians, physicists, and historians of group actions on manifolds. Lie regarded such mathematics. He further notes that from his own transformations, the elements of which form what experience “the understanding of a theory is deep- is now called a local , as generated by the ened by familiarity both with the considerations infinitesimal maps associated with the first deriv- that motivated various developments and with the atives. To pass from these infinitesimal genera- less formal, more intuitive manner in which they tors to the local transformations required finding were initially conceived.” By implication this means the solutions to a system of differential equations, that readers should have a reasonably good un- and it was this idea that lay at the heart of Lie’s derstanding of modern Lie theory; for those who vision. From Jacobi’s work Lie also recognized that do not, Hawkins recommends a number of standard the elements of the “infinitesimal group” satisfied texts on the subject. Clearly the mathematical the familiar bracket relations for a Lie , but demands he places on his readership are very high, it was Wilhelm Killing who first explored their and he does not flinch from presenting detailed structure theory as part of a program aimed at technical ideas throughout. classifying space forms. As for the global theory Still, if knowledge of modern Lie theory is helpful, of Lie groups, not even an inkling of such a possi- even essential, to understand parts of this study, bility existed before the 1920s. Thus, only in the another prerequisite seems to me even more final chapters will readers begin to make contact crucial, namely, an openness to and appreciation with some of the more familiar ideas of modern Lie for the often primitive and usually unfamiliar mathematical ideas of the past. Historians of 1Quoted on the dust jacket of Arild Stubhaug, The mathematics take such an attitude for granted, but Mathematician Sophus Lie, Springer-Verlag, 2002.

JUNE/JULY 2003 NOTICES OF THE AMS 669 theory. With these general remarks in mind, let beginning with Sophus Lie’s own ideas and those me now turn to take a closer look at Hawkins’s inspired by his work. In Chapter 1 the reader en- study. counters what might seem like a rather bewilder- ing chain of unfamiliar concepts: tetrahedral com- Origins of Lie Theory and the Erlangen plexes, W -curves and W -surfaces, and Lie’s Program line-to- mapping. Lie was absorbed by the In the preface the author notes his indebtedness remarkable properties of his line-to-sphere mapping to other historical studies, but he also emphasizes during his sojourn in Paris with the originality of his in 1870. Klein later described how Lie’s mind lived analyses of eight partic- in these spaces of lines and , flip-flopping ular topics to which he to and fro. Eventually, the properties he uncovered attaches special signifi- by means of these obscure mental gymnastics led cance: (1) the roots of him to formulate a general geometric theory of Lie’s theory in geomet- contact transformations, a notion that found its way rical and analytical prob- into Klein’s “Erlangen Program” in 1872. lems, several taken up All these ideas and many more are part of together with Felix Hawkins’s account in Chapters 1 and 2, in which Klein; (2) Killing’s work he describes the origins of Lie’s theory of contin- on foundations of geom- uous groups. According to Lie himself, this theory etry as an expression of was born in the winter of 1873–74, and Hawkins Weierstrassian research carefully elucidates what happened during the ideals; (3) how the study preceding period, the years 1869–1873, by divid- of space forms led to ing Lie’s early career into two subperiods. Up Killing’s work on the until the fall of 1872 he worked closely with Klein, structure of Lie algebras; but afterward Lie went his own way, beginning an (4) Cartan’s revamping intensive study of systems of partial differential of Killing’s faulty theory equations inspired especially by Jacobi’s analytical of secondary roots; (5) ideas. Hawkins sets out this periodization at the Sophus Lie, circa 1870. how research ideals in beginning of Chapter 1, but without mentioning Paris affected the re- the landmark work that came at the end of the ception of Lie’s theory, including Cartan’s work on Klein-Lie collaboration, the Erlangen Program. Lie algebras; (6) an analysis of work undertaken in Instead, he reserves this topic for the final section Lie’s school on representations of Lie algebras; (7) of the chapter, at the close of which he emphasizes a view of Weyl as the leading exponent of a par- that Lie saw the contents of Klein’s Erlangen ticular mathematical style cultivated in Hilbert’s Program as only tangentially related to his own Göttingen; (8) how Weyl’s work on general relativ- program for classifying all continuous groups. ity paved the way for his investigations of the Hawkins’s periodization is both clear and con- structure theory of Lie groups. The first three vincing, but unfortunately his handling of this themes will be familiar to those who have followed topic tends to sweep a whole batch of problems Hawkins’s earlier work, to which he refers the under the carpet, beginning with the controversial reader in several places in the first two parts of the status of Klein’s Erlangen Program. To grasp what present study. Roughly two-thirds of its contents, was at stake, the reader ought to be told early on Parts III and IV, take up the latter five topics, which about Lie’s ambivalent views regarding the Erlan- are substantially new. All eight of these themes have gen Program, one of several contentious issues considerable significance for Hawkins’s overall that caused his relationship with Klein to sour story. Unfortunately, the reader has to invest a during the early 1890s. Given the importance of the great deal of time and patience to see why. Perhaps Lie-Klein partnership for Hawkins’s whole story, the author thought the book long enough already, one would have hoped to learn more about their but he surely could have made it more accessible sometimes volatile encounters as well as their had he chosen to sketch the overall contours of his respective mathematical styles. Both were geome- argument in a lengthier introduction rather than ters who relied heavily on intuitive ideas, but Klein relying mainly on general summarizing remarks tended to think in global terms, whereas Lie fixed made along the way, helpful as these certainly are. his attention on differential properties, ignoring Hawkins focuses primarily on mathematical ideas most global issues. Hawkins underscores the fact and intellectual contexts, with rather scant (occa- that Lie’s program failed to distinguish sharply sionally too scant) attention paid to interactions be- between the local and global theory of Lie groups. tween the personalities involved. Part I thus exam- Lie often tended to regard the latter as generated ines the rich intellectual framework that led to the by the former in a straightforward manner (as in modern theory of Lie groups and Lie algebras, the case of the commutative one-parameter groups

670 NOTICES OF THE AMS VOLUME 50, NUMBER 6 associated with W-curves that he studied with that standard myths about Klein’s Erlangen Pro- Klein). Klein’s Erlangen Program, on the other hand, gram have continued to live on despite the fact was exclusively concerned with transformation that Hawkins exploded many of these long ago. In groups that act globally on a manifold. As Hawkins Chapter 2 he presents an abridged recapitulation duly notes, Klein’s whole outlook was deeply of his main findings regarding Lie’s path of rooted in projective methods, yet unlike Lie, he discovery after he and Klein parted ways. Here emphasized the distinction between global and Hawkins carefully chronicles the crucial events local properties of manifolds early on and often. from 1873 to 1874 that led Lie to his theory of As already mentioned, these key insights might continuous groups, an interest that had not yet have been spelled out more effectively in an formed when he consulted with Klein about the introductory essay that surveyed some of the book’s ideas that went into the Erlangen Program. major themes before exposing the reader to the morass of technical information Hawkins draws Appeasing the Analysts upon in making his arguments. With regard to the In Chapter 3 Hawkins presents the key elements Erlangen Program, for example, it should be em- of Lie’s theory as well as the main results he ob- phasized that Klein did not originally conceive it tained during the period 1874–1893. A few words as a research program for continuous groups but of caution are needed here. For the most part rather as a proposal to study systematically the Hawkins deals with Lie’s theory as it appeared in invariants and covariants of known groups. It has the three volumes entitled Theorie der Transfor- seldom been noticed that neither he nor Lie chose mationsgruppen rather than citing Lie’s earlier to pursue this core challenge directly after 1872, papers. He points out that these volumes were whereas Eduard Study, who enters Hawkins’s story written with the assistance of Friedrich Engel and in a number of prominent places, did so beginning published between 1888 and 1893. Although Engel in the mid-1880s. Study’s work, however, was comes up in several other places in this study, in quickly forgotten, whereas Klein successfully pro- particular as Killing’s correspondent, Hawkins moted the oft-repeated story that he and Lie had skirts around the messy problems that developed already planned to divide between between Lie and Engel, his principal disciple, as a them back in 1872. result of that correspondence. At least one aspect In the preface to his 1884 book on the icosahe- of their rocky relationship should have been ad- dron, Klein wrote that he and Lie decided to “di- dressed directly, however: namely, the extent to vide and conquer” group theory after they parted which these three volumes adequately reflect ways in 1872, Klein taking the theory of discrete Lie’s own thinking. Hawkins briefly recalls the kind groups and automorphic functions, while his for- of working relationship Lie developed with his mer collaborator took on the even harder theory new-found disciple when Engel came to Norway in of continuous groups and their related differential 1884, citing Engel’s own recollections. According equations. By the 1890s this misleading story had to these, they met twice daily so that Lie could become thickly entwined with the then-emerging outline the contents of the individual chapters, “a mythology surrounding the Erlangen Program. sort of skeleton, to be clothed by me in flesh and Later, taking up where Einstein and Minkowski left blood” (p. 77). Hawkins further notes that the man- off, Klein successfully promoted the idea that his uscript went through many revisions and that the Erlangen Program actually presaged the mathe- whole project led to a text five times longer than matics of relativity theory, which he interpreted as originally planned. Still, he does not pursue a key the invariant theory of the . By the issue raised by Hans Freudenthal, who gave this early 1920s thousands had read about the signif- assessment in a biographical essay on Lie: icance of Klein’s Erlangen Program in Oswald Spengler’s Decline of the West. Thus, in the course The nineteenth-century mathematical of fifty years this initially obscure expository essay public often could not understand lucid had been elevated to the status of a modern clas- abstract ideas if they were not expressed sic. Presumably only a few commentators studied in the analytic language of that time, it closely, but this did not prevent several distin- even if this language would not help guished mathematicians (many with close ties to to make things clearer. So Lie, a poor Klein) from writing about the significance of the analyst in comparison with his ablest Erlangen Program. Heroic feats have occupied a contemporaries, had to adapt and major place in mathematical lore ever since Plutarch express in a host of formulas, ideas wrote about how Archimedes single-handedly held which would have been better without off the Roman legions who besieged his beloved them. It was Lie’s misfortune that by Syracuse. Debunking historical myths, on the other yielding to this urge, he rendered his hand, has rarely captivated mathematicians’ obscure to the geometricians attention. Thus, it should not come as a surprise and failed to convince the analysts (Hans

JUNE/JULY 2003 NOTICES OF THE AMS 671 Freudenthal, “Marius Sophus Lie”, well known argument regarding the wider method- Dictionary of Scientific Biography [New ological significance of Weierstrass’s theory of ele- York: Scribners, 1973], 8:325). mentary divisors. Within the Berlin school, rigor in algebra meant undertaking an exhaustive analysis of Engel, the “ghostwriter” of Lie’s three volumes a problem, and for this purpose the theory of ele- on the theory of transformation groups, must have mentary divisors proved to be a crucially important had a major hand in this process, aimed at winning tool that enabled algebraists to go beyond “generic the recognition of leading contemporaries. Initially arguments”, usually based on counting coefficients, at least, these “analysts” were, first and foremost, that held only for the “general case”. Such arguments the mathematicians closely associated with Weier- were particularly common in enumerative geome- strass’s school in Berlin, several of whose members try, a field in which controversies over standards of play major roles in Hawkins’s book. Although he rigor abounded during the nineteenth century. As discusses the misgivings Weierstrass and company an exponent of the new Berlin ethos, Killing demon- had about Lie’s work, Hawkins never hints that strated in his early work on space forms a deep com- Engel might have played a significant role in the mitment to exploring not just the general cases but dilemma Freudenthal described. Lie felt isolated in all the possible degenerate ones as well. In the clos- Norway, and he was frustrated over the difficulties ing section of Chapter 4, Hawkins contrasts Killing’s he encountered in trying to find an audience for approach with Klein’s tendency to shunt such com- his work. Klein and Adolf Mayer therefore sent plexities aside or, better yet, relegate them to one of Friedrich Engel to aid him, and it seems likely that his doctoral students. In the foundations of geome- Engel thought Lie’s mathematics had to be made try Klein was long content to explore only the stan- more palatable for analysts. This interpretation dard Euclidean and non-Euclidean models, despite gains credibility from the testimony found in a the fact that already back in 1873 W. K. Clifford had footnote on page 190. There Hawkins cites the au- made him aware of topological differences that can thority of Lie’s former student Gerhard Kowalewski, arise when considering manifolds of constant cur- who claimed that Lie never referred to the three vol- vature. By placing Killing’s work within the context umes written by Engel, with their “function- of Berlin’s research tradition in algebra, Hawkins theoretic touch”, but rather always cited his own adds a significant new dimension to his story, for papers when discussing Lie was essentially a geometer turned analyst, but his work with others. If most decidedly not an algebraist of Killing’s caliber. so, one might well won- Moreover, after Cartan picked up the pieces of der whether Lie even Killing’s work, two other leading Berlin algebraists, knew the contents of his Georg Frobenius and Isaai Schur, emerged to play own “masterpiece” all that well! Taking Freudenthal’s key roles in the story leading up to Weyl’s break- argument a step further, through in the 1920s. perhaps Lie’s real mistake In Chapter 5 Hawkins carefully describes Killing’s was letting Engel dress up ensuing research program, which took its point of his geometrically inspired departure in the Riemann-Helmholtz space prob- ideas in analytic garb lem. This naturally led Killing to a systematic (though Mayer had al- analysis of r-dimensional Lie algebras based on ready been pushing Lie in the degree of mobility attributed to rigid bodies this direction before Engel in an n-dimensional space. The background for stepped onto the scene). this program was sketched in Killing’s 1884 essay, At any rate, by relegating “The Concept of Space Extended”, but Hawkins Engel to the background conjectures that by this time Killing already had in Chapter 3, Hawkins developed the approach and methods he took in leaves these interesting his four major papers from 1888 to 1890 on the Wilhelm Killing, circa 1890. questions unexamined. structure of finite-dimensional simple Lie algebras over the complex numbers. Hawkins arrived at this conclusion not only by comparing the earlier Rivalry of Lie and Killing essay with Killing’s later work but also through In Chapter 4 Wilhelm Killing makes his entrance as careful analysis of Killing’s correspondence with a rather different kind of exponent of the Berlin math- Engel, which began in late 1885 after the latter ematical milieu. Killing was preoccupied mainly with had returned to Leipzig from Norway. Although problems stemming from algebra and foundations of he describes this correspondence in some detail, rather than ; his preferred Hawkins neglects to say anything about its wider brand of Weierstrassian rigor thus had more to do repercussions, alluded to above. with the theory of elementary divisors than conven- Initially, no apparent signs of conflict arose, and tional epsilontics. Here Hawkins presents his by now in the summer of 1886, shortly after Lie’s arrival

672 NOTICES OF THE AMS VOLUME 50, NUMBER 6 as Klein’s successor in Leipzig, Killing met with Lie comes closer to the truth. I value Klein’s and Engel. Lie must have known all along that Engel talent highly and will never forget the had been writing to Killing, but he apparently at first sympathetic interest with which he saw the latter as a friendly rival whose work would has always followed my scientific only enhance the stature of his theory. He changed endeavors. But I do not feel that he has his mind, however, in early 1888 when he saw a satisfactory understanding of the the first installment of Killing’s four-part study difference between induction and and wrote immediately to Klein: “Mr. Killing’s work proof, or between a concept and its in Mathematische Annalen is a gross outrage application (Stubhaug, p. 371). against me, and I hold Engel responsible. He has Not surprisingly, these remarks scandalized certainly also worked on the proof corrections” (Stubhaug, p. 368). Lie now reached the conclusion many in the German mathematical community (or that too many of his ideas had been communicated at least outside Berlin). By publicly belittling Klein, to Killing by Engel, ideas Lie regarded as his ex- Lie not only put an end to their friendship but also clusive intellectual property. In Lie’s eyes, Engel effectively severed his ties with nearly all his for- had betrayed his trust, and their relationship never mer allies in Germany. Probably he felt he had to fully recovered from this bitter episode. The fol- take such a radical step to free himself from Klein lowing year Lie had to be placed in a psychological if he were ever going to strike out on his own. At clinic, as he could no longer sleep at night. His wife any rate, this was no mere isolated outburst, but brought him home to Leipzig in the summer of rather part of a long-term shift of allegiances, as 1890, but his condition did not improve until many Lie continued to cultivate ever-stronger ties with months later (for details see Stubhaug, pp. 350–9). the French mathematical community. Hawkins is undoubtedly right that Killing’s work Cartan and the French Reception owed little to Lie’s ideas: not only was it motivated by an independent research agenda but Killing uti- Hawkins takes up this very theme at the beginning lized techniques that lay outside Lie’s repertory. In of Chapter 6 as background for his discussion of this sense the misunderstandings that developed Cartan’s thesis from 1894 on semisimple Lie alge- between Lie and Killing had no direct bearing on bras. One should note, however, that Lie’s renewed their respective contributions to the theory. Yet interest in the reactions of the French community these unfortunate developments were clearly re- went hand-in-hand with growing disillusionment lated to Lie’s growing sense of disillusionment over with the reception of his work in the German the reception his ideas had received within the mathematical world. Lie sought recognition for German mathematical community. By the early his theory, but he was not content with the kind of 1890s Lie was most unhappy about many things support he got from the likes of Engel and Study, in Leipzig, some of them purely personal, but oth- who were marginal figures in the German commu- ers directly related to his standing among German nity. Hawkins explains why the situation was, in mathematicians. Since arranging Lie’s appointment general, much more favorable in France. Gaston in 1886, Klein hoped to forge an alliance with him, Darboux had shown an early interest in Lie’s work, a new Göttingen-Leipzig axis, against the Berlin and in 1888 he encouraged two graduates of the establishment. But the Norwegian wanted to avoid École Normale, Vladimir de Tannenberg and Ernst tying himself too closely to Klein’s network of Vessiot, to study with Lie in Leipzig. Vessiot, power. Eager to build up his own school in Leipzig, following the lead of Émile Picard, took up Lie’s he viewed potential competitors with suspicion. Re- original vision, namely, to develop a garding Lie’s attitude toward Killing’s work, of differential equations. In fact, nearly all the Hawkins mentions only the positive remarks Lie French mathematicians were primarily interested made in 1890 (p. 172), ignoring the highly critical in applications of Lie’s theory, not the structure ones from before and afterward (see Stubhaug, theory itself. As Hawkins shows, even Élie Cartan pp. 382–5). shared this viewpoint to some extent. These circumstances strongly colored the last Intellectually, Cartan’s work was directly linked phase of Lie’s career, and all his writings after to Killing’s, and yet the spirit that guided the 1892 (several of which Hawkins cites in various Frenchman differed markedly from the ethos that places) should be read with this background in inspired Weierstrass’s pupil, Killing. In Chapter 6 mind. In the preface to the third volume of his The- Hawkins asks why it took Cartan so long to return orie der Transformationsgruppen, published in to Killing’s original program aimed at determining 1893, Lie attacked several mathematicians at once, the irreducible representations of all semisimple but most notably Felix Klein, about whom he wrote: Lie algebras. The answer, he suggests, can be found by examining typical attitudes of the Parisian I am no pupil of Klein’s. Nor is the re- community toward Lie’s theory as it became in- verse the case, even though it perhaps creasingly familiar to them throughout the late 1880s

JUNE/JULY 2003 NOTICES OF THE AMS 673 and early 1890s. What mathematicians like Poincaré , Italian-style. The remainder of and Picard valued most about Lie’s theory of con- the chapter deals with technical issues: Cayley’s tinuous groups was its potential applications to counting problem in the theory of algebraic forms geometry and differential equations. On the other and their associated invariants, and a subsequent hand, they showed far less interest in the structure innovation by Kowalewski involving Cayley’s notion theory and its related problems. Hawkins contrasts of weights that enabled Kowalewski to find a few the open-minded views of leading Parisian mathe- projective groups that leave nothing planar in- maticians with the sharp rejection voiced by Frobe- variant. In Chapter 8 Hawkins pulls all these tech- nius, who became Berlin’s leading mathematician nicalities together, showing how these intermedi- after Weierstrass retired in 1892. The latter had ear- ate developments permeated Cartan’s trilogy of lier confided privately that he papers. By this point, the interplay of diverse math- considered Lie’s work, pre- ematical ideas becomes almost dizzying, making sumably in the form pre- the author’s mastery of the material quite won- sented by Engel in Theorie der derful to behold. Transformationsgruppen, so wobbly that it would have to Weyl and Göttingen Mathematics be reworked from the ground Still, Hawkins saves the best for last in four chap- up. Frobenius went even fur- ters centered on Hermann Weyl that bring his study ther, claiming that even if it to a brilliant close. In Chapter 9 he deals with the could be made into a rigor- context of Weyl’s early career in Göttingen when ous theory, Lie’s approach to he was closely associated with and differential equations repre- his school. Thus, Hawkins begins with an account sented a retrograde step com- of Hilbert’s work in three fields of central impor- pared with the more natural tance for the story: invariant theory, integral equa- and elegant techniques for tions, and . In his early work solving differential equations on integral equations, Weyl extended the results of developed by Euler and La- Hilbert and Hellinger to cases involving singular ker- grange. Needless to say, the nels, and in his Habilitationsschrift from 1910 he

Photograph courtesy of Henri Cartan. leading French mathemati- developed the spectral theory for second-order lin- Élie Cartan, 1903. cians felt otherwise. Among ear differential equations with singular boundary the younger generation, Car- conditions, thereby generalizing classical Sturm- tan showed the strongest Liouville theory. Weyl was deeply influenced by affinity for the abstract problems associated with Hilbert’s ideas and shared his mentor’s universal- Lie’s theory. Yet, as Hawkins convincingly argues, ism. This early work carried many of the earmarks even Cartan was not tempted to go beyond that por- of Hilbert’s Göttingen school, whose practitioners tion of Killing’s program deemed relevant for the were mainly inspired by concrete problems and theory of differential equations, the remainder hav- potential physical applications (in particular elas- ing only “artistic value”, a term Picard and Poincaré ticity theory) rather than the challenge of devel- used disparagingly. oping a general theory of function spaces. The true In Chapter 7 Hawkins describes various transi- pioneers of abstract were out- tional developments, many involving ideas that siders like Maurice Fréchet, Friedrich Riesz, and were significant for Cartan’s trilogy of papers on Ernst Fischer. Weyl was the consummate insider. semisimple Lie algebras from 1913 to 1914, the Hawkins gives an overview of Weyl’s mathemati- main topic he takes up in Chapter 8. By the 1890s cal career up to 1913, when he published Die Idee Lie had managed to build up an important school der Riemannschen Fläche. He then comes full specializing in various aspects of his vast research circle, closing Chapter 9 with a discussion of program. Hawkins focuses particularly on those Hilbert’s brand of mathematical thinking. Hawkins works bearing on linear representations of Lie al- sees this, in essence, as a manifestation of what gebras. Once again, Engel and Study enter the pic- Hilbert once called “Riemann’s principle”, accord- ture, although the latter’s work on projective groups ing to which proofs should be driven not by cal- never really came to fruition. By the early 1890s culation but solely by ideas. Personally, I think it Klein’s Erlangen Program also became widely known would have been better not to resurrect this ter- for the first time, as it was reissued in numerous minology. After all, Minkowski said essentially the translations. The first of these, fittingly enough, same thing about Riemann’s teacher, calling this was produced in Italy, soon to emerge as the lead- tendency the “true Dirichlet principle”. ing nation for geometrical research. Hawkins deftly Obviously Weyl was heavily indebted to Hilbert, characterizes the work of Corrado Segre and his stu- as Hawkins aptly describes, but one should not dent, Gino Fano, both of whom pushed Klein’s ideas overlook other important influences. In his book- into the forefront of research in higher-dimensional let on Riemann surfaces, Weyl emphasized the

674 NOTICES OF THE AMS VOLUME 50, NUMBER 6 importance he attached to Klein’s essay from 1881 Isaai Schur, exerted a deep and lasting impact on in which the of the surface determines the the theory of Lie groups. periodicity of the moduli directly. Weyl looked to Picking up on Frobenius’s theory of group char- Einstein for inspiration in physics and Brouwer for acters and an 1894 paper on invariant theory by a vision of the continuum. Indeed, these intuitive Adolf Hurwitz, Schur tackled the problem of thinkers had a lasting impact on Weyl, who criti- finding a wide class of representations of GL(n, C). cized the general tendency to sacrifice content for In the course of doing so, he found continuous form in mathematics, a view that flew in the face analogues to several of Frobenius’s results for of Hilbert’s strongly held position on the founda- finite groups and presented these in his 1901 dis- tions of mathematics. In 1921 Weyl made his sertation. His supervisor heaped lavish praise on opposition to Hilbert’s views known publicly by this work, calling Schur “a master of algebraic announcing the “new foundations crisis” in math- research” (p. 402), but oddly enough Frobenius did ematics, a controversy that pitted Hilbert’s for- nothing to encourage his pupil to publish it in a malism against Brouwer’s intuitionism. More than mathematical journal. Although Hawkins devotes two decades later, in his obituary for Hilbert, Weyl considerable space to this pioneering study, he once again distanced himself from his former men- also emphasizes that few were familiar with tor’s views on axiomatics, which he saw as the Schur’s results until the early 1920s. In the mean- weakest part of Hilbert’s legacy; what he admired time, Schur devoted most of his efforts to finite most were his contributions to . All groups, algebraic forms, and their associated in- this suggests that Weyl’s intellectual indebtedness variants, topics Hawkins discusses in the remain- to Hilbert was anything but simple, whereas ing sections of Chapter 10. Hawkins’s picture, based on their shared propen- There he portrays Schur’s work as a continuation sity for ideas rather than technical arguments (Rie- of the Berlin tradition in algebra. He also argues that Schur, like Frobenius, did not “look to the schools of mann’s principle) overlooks too many other im- Klein or Lie for significant mathematics” (p. 404). portant factors, including intellectual tensions. As Perhaps this is true, though one should remember Hawkins’s study demonstrates so well, Weyl never that Hurwitz had been Klein’s star pupil. But more aligned himself with a school or a particular importantly, we should be wary of attributing Frobe- methodological approach; he was always turning nius’s polemical and often narrow-minded views to to new and eclectic sources. Ultimately and perhaps Schur, who seems to have kept his distance from the ironically, his work on the representation theory of running ideological debates that preoccupied his Lie groups owed more to the Berlin algebraic tra- Doktorvater. Hawkins cites no provocative remarks dition than to Hilbert’s ideas. from Schur’s pen. Instead, he shows how Hurwitz’s Work of Frobenius and Schur ideas inspired Schur’s work and how he later col- laborated with Alexander Ostrowski and Weyl, both Hawkins lays the groundwork for this theme in from the Göttingen camp. Weyl’s work built directly Chapter 10, which deals with the contributions of on Schur’s, but he also drew on another important Frobenius and Schur. He also picks up some of the source of inspiration that Hawkins discusses here. topics introduced in Chapter 7 (Cayley’s counting In 1897 Adolf Hurwitz introduced a new method of problem and Kowalewski’s theorem). After a de- integration in order to derive the invariants for tailed account of Frobenius’s theory of group char- various continuous groups. Hurwitz devised a tech- acters and representations, Hawkins offers some nique, later dubbed by Weyl “the unitarian trick”, to speculations as to why Berlin’s leading algebraist carry out the integration process on the bounded probably never contemplated an extension of his subgroup of unitary transformations of SL(n, C) . theory to continuous groups like the one Weyl Hurwitz’s methods lacked rigor, since they relied on achieved afterward. He suggests that Frobenius’s Lie’s theory, but as Hawkins points out there was no animus against Lie’s theory was so strong that he other theory that dealt with global issues. (It was not probably wrote off work he thought was contam- until 1933 that Alfred Haar developed a rigorous inated by Lie’s ideas. Thus, Hawkins found no approach, one that also showed how the theory of evidence that Frobenius ever attempted to study Peter-Weyl could be extended to compact topologi- Cartan’s work on semisimple Lie algebras, pre- cal groups.) sumably for this very reason. Moreover, by the time he completed his main work on group char- Weyl’s Journey from Relativity to acters and representations in 1903, Frobenius Representations found himself caught in a losing battle with Klein, In the final two chapters Hawkins describes Weyl’s who by now had both Hilbert and Minkowski at his intellectual journey from relativity to representa- side in Göttingen. Yet even if Berlin could no longer tions, concluding with an analysis of his seminal compete with Göttingen in the realm of academic papers from 1925 to 1926. Several elements of politics, the work of Frobenius and his star pupil, this story can be found in Armand Borel’s lecture

JUNE/JULY 2003 NOTICES OF THE AMS 675 “Hermann Weyl and Lie Groups”, delivered at Matter, Weyl retained this theory despite Einstein’s the Weyl Centenary Celebration in 19852 (a text vigorous opposition. But by the time he prepared Hawkins cites with praise in his suggestions for the fifth, he no longer had any faith that a pure field further reading). Aside from this essay, though, theory could capture the intricate properties of Hawkins had to rely almost exclusively on published matter. papers and some scant though suggestive hints Still, Weyl did realize that the time was ripe to from Weyl’s later writings in order to pull together reconsider another important physical issue in the a coherent picture. One of these hints was dropped light of general relativity, namely, the properties of in Weyl’s 1949 article “Relativity Theory as a Stim- physical space. Since Einstein’s theory undermined ulus to Mathematical Research”, in which he wrote: the classical notion of a rigid body, the funda- “…for myself I can say that mental starting point for the original Riemann- the wish to understand what Helmholtz-Lie space problem, Weyl formulated a really is the mathematical new set of spatial properties and showed that his substance behind the formal version of the space problem reduced to finding the apparatus of relativity the- orthogonal Lie algebras associated with a nonde- ory led me to the study of generate . As Hawkins relates, this representations and invari- work from 1921 to 1922 brought Weyl into direct ants of groups.” Borel began contact with Cartan’s earlier deep investigations. his lecture with this telling He goes on to trace Weyl’s growing interest in the quote, but Hawkins provides properties of tensors, a topic directly the first detailed look at the linked to his earlier work on general relativity and story behind Weyl’s words. unified field theory. As was well known, the In 1916 Einstein unveiled Riemann-Christoffel tensor satisfies several sym- his mature theory of gravi- metry conditions that reduce its 256 components tation in a now classic paper to just 20 algebraically independent entities. Weyl that began by laying out the was among the first, however, to tackle the general formal apparatus of the gen- problem of classifying tensors by means of their Photograph by RIchard Courant, used with permission of eral theory of relativity: the (another was J. A. Schouten, whose Ernst Courant. techniques of Ricci’s ab- work Weyl detested for its “orgies of formalism Hermann Weyl (right) with solute differential , which Einstein had learned …[that] threaten the peace of even the technical sci- David Hilbert, circa entist” [Space. Time. Matter, p. 54]). mid-1920s. from his friend Marcel Gross- mann. As Hawkins recounts, Weyl had strong views about how to do mathe- Grossmann was familiar with matics and how it should best be presented. These the lengthy 1901 article in Mathematische Annalen come into view in Hawkins’s story by way of Weyl’s that Ricci and his student Tullio Levi-Civita had writ- response to criticism from Eduard Study, who ten at Klein’s request. Whether or not Weyl knew claimed in the preface of his book on the theory the Ricci calculus remains unclear, but he quickly of invariants from 1923 that the author of Raum. threw himself headlong into Einstein’s theory, and Zeit. Materie was “behind the times, and not just a during the summer semester of 1917 he began lec- little” with regard to the modern theory of group turing on general relativity at the Eidgenössische invariants. Study felt that Weyl had overplayed the Technische Hochschule. At the suggestion of Ein- importance of the Ricci calculus while ignoring stein’s friend, Michele Besso, he published the re- Study’s own forte, the symbolical method for gen- sults the following year in book form as Raum. Zeit. erating invariants devised by Clebsch and Aronhold. Materie, one of the enduring classics of relativity Weyl answered Study’s charge with some polemi- theory. As Erhard Scholz has shown, a substantial cal remarks of his own, but Hawkins shows that he part of Weyl’s intellectual journey can be traced also went far beyond this initial response. Indeed, through the four different editions of Raum. Zeit. his subsequent research on tensor algebra led him Materie that were published by Springer between to Frobenius’s theory of group characters, which 1918 and 1923 (the first two editions were virtu- he utilized as a tool for determining the scope of ally identical). The third edition from 1919 incor- the symbolical method Study loved so much. porated his unified field theory for gravitation and Hawkins even tries to capture Weyl’s state of mind electrodynamics based on a generalization of Rie- at the time, a perhaps dubious endeavor that gains mannian geometry. In the fourth edition of 1921, credibility, however, in light of Weyl’s later as- familiar from the English translation, Space. Time. sessment: “Here those problems which according to Study’s complaint the relativists had let go by 2Armand Borel, “Hermann Weyl and Lie Groups”, Hermann the board are attacked on a much deeper level Weyl 1885–1955 (K. Chandrasekharan, ed.), Springer- than the formalistically minded Study had ever Verlag, New York, 1986, pp. 655–85. dreamt of” (p. 455).

676 NOTICES OF THE AMS VOLUME 50, NUMBER 6 Emergence of Modern Lie Theory way. Hawkins’s account of this strange but won- The bridge from relativity to representations of derful saga resurrects a heroic chapter in the Lie groups having been crossed in 1923–24, Weyl history of mathematics. For anyone with a serious took up the general problem of developing a interest in the rich background developments that global theory for the latter. In Chapter 12 Hawkins led to modern Lie theory, this book should be offers a lengthy analysis of Weyl’s 1925 paper browsed, read, savored, and read again. “Theory of the Representation of Continuous Semi-Simple Groups by Linear Transformations”. Following this, he takes up Cartan’s response to this work, ending with a discussion of the joint paper The photographs on pages 670, 672, and by Weyl and Fritz Peter from 1926. Drawing on the 676 of this article are reprinted from Emergence numerous elements laid out in earlier chapters, of the Theory of Lie Groups with permission Hawkins nicely reveals the rich threads that Weyl of Springer-Verlag; Thomas Hawkins; the wove into the fabric of his theory. Soon thereafter Department of Mathematics, University of Oslo; Cartan modified some of this work, bringing out Universitäts- und Landesbibliothek Münster; its more topological features. A more abstract and Ernst Courant. The photograph on page approach to universal covering groups was intro- 674 was lent to the Notices by Henri Cartan. duced by Otto Schreier in 1925, but neither Weyl nor Cartan was aware of it at the time. The Peter- Weyl paper laid the groundwork for applications to functional analysis by creating a generalized theory of Fourier series for compact groups that admit a translation-invariant integration process akin to the one Hurwitz had found earlier. This opened the way for general Haar measures and other developments, including von Neumann’s work on topological groups. As Hawkins notes, Weyl took no direct part in these subsequent developments, though he did make an important contribution to Harald Bohr’s theory of almost periodic functions. He then ends with a brief afterword mainly concerned with suggestions for further reading. This book is anything but easy reading, owing mainly to the mathematical and historical com- plexities involved. Yet whatever its weaknesses and however these might be judged, this study is just as clearly a stunning achievement. Few histo- rians of mathematics have made a serious attempt to cross the bridge joining the nineteenth and twentieth centuries, and those who have made the journey have tended to avert their eyes from the mainstream traffic. Most such studies have focused on new breakthroughs in foundations, , , and topology, whereas only scant attention has been paid to develop- ments rooted in earlier work, like Lie’s theory of continuous groups. In my eyes the single greatest merit of Hawkins’s book is that the author tries to place the reader smack in the middle of the action, offering a close-up look at how mathematics gets made. Of course, this story has a happy ending, but what Hawkins shows is the surprising diversity of approaches to and motivations for this work, how it moved forward in unpredictable fits and starts, with periods of buzzing activity separated by in- terludes of quiet stagnation. No royal roads, indeed, though there was plenty of inspiration along the

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